1 Introduction and Main Results
In this work we consider the problem of finding solutions to
(1.1)
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
+
g
(
u
)
=
μ
on
∂
ℝ
+
N
,
where 1<p≤N, N≥2, ℝ+N={(x′,xN):x′∈ℝN-1,xN>0}, μ∈𝔐b(∂ℝ+N). Here 𝔐b(∂ℝ+N) is the space of Radon measures in ℝN with bounded total variation which are supported in ∂ℝ+N={(x′,xN):x′∈ℝN-1,xN=0}, uν is the normal derivative of u, g:ℝ→ℝ is a continuous function, and
-
Δ
p
u
:=
-
div
(
|
∇
u
|
p
-
2
∇
u
)
.
Consider the related problem of finding a solution to
(1.2)
{
-
Δ
p
u
+
g
(
x
,
u
)
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a bounded domain in ℝN, μ∈𝔐b(Ω), and g:ℝN×ℝ→ℝ is a continuous function, nondecreasing in the second variable. If p>N, a unique solution can be obtained by the theory of monotone operators from W01,p(Ω) into its dual W-1,p′(Ω), since in this case any bounded measure in Ω belongs to this dual.
When 1<p≤N, the study of problem (1.2) is based upon the theory of renormalized solutions. In the case g≡0 the concept of renormalized solution, for general measure data, was first introduced in [8], wherein the authors showed existence and partial uniqueness results. The proof of existence relies in a delicate and very technical stability result. The concept of renormalized solution has been since then the main tool to study degenerate elliptic problems with measure data. We refer the reader to [23] for an overview of the concept and further references.
When g is nontrivial, the nature of equation (1.2) depends on the sign of g(x,u)u. In accordance with the literature, we call it a problem with absorption when g(x,u)u≥0, and a problem with source when g(x,u)u≤0. Further, we say that the problem is subcritical if it can be solved for any bounded measure. If it is not the case, we say that the problem is supercritical, and in that case conditions on the measure expressed in terms of concentration with respect to some capacities are needed for solving the problem.
For the problem with absorption, existence of renormalized solutions to (1.2) in subcritical cases has been shown in [3] and [23]. In [3] the author considers g(u)=|u|q-1u, while more general nonlinearities are considered in [23]. Let us mention that in the power case one obtains the sufficient condition q∈(0,N(p-1)N-p) if 1<p<N, and q≥0 if p=N. In the case p=N exponential-type nonlinearities are considered, but under a restriction in the size of the measures.
The problem with absorption in supercritical cases has been studied in [4]. There the authors show that given a fixed nonlinear term g(x,s) existence of renormalized solutions to (1.2) holds for a certain class of measures. Their results, which are quite general, are based mainly on a delicate study of the Wolff potential and can be applied to establish sufficient conditions for existence of renormalized solutions when the form of the nonlinearity is more explicit. For example, when g=|u|q-1u, q>p-1, a sufficient conditions on the measure is that it be absolutely continuous with respect to the Lp,qq-p+1(ℝN) Bessel capacity.
Problem (1.2) is more difficult when it is of source type. In fact, in this case only nonnegative solutions are considered. In [16] the authors show equivalent sufficient conditions to obtain nonnegative renormalized solutions to (1.2) when g(u)=-uq and μ is a nonnegative measure. It is also shown that, if the measure is compactly supported in Ω, these conditions are necessary. Their results, which are obtained through a careful study of the Wolff potential and its relationship to the Bessel and Riesz potential, show in particular that if q∈(p-1,N(p-1)N-p) and 1<p<N, or q≥p-1 and p=N (i.e., the subcritical case), then any nonnegative measure with small enough norm admits a solution. In the supercritical case, a sufficient condition is that the measure must be “Lipschitz” continuous with respect to the Lp,qq-p+1(ℝN) Bessel capacity. We note also that their main results allow them to present a complete characterization of removable sets for (1.2) in terms of some fractional Bessel capacities, as well as to prove Liouville-type results for problems in the whole ℝN.
Going back to problem (1.1), in the case p>N we can similarly try to obtain a unique solution in the space W1,p(ℝ+N) by using the theory of monotone operators. Indeed, this approach has been successfully applied in [24] to a Neumann-type subcritical problem with absorption, in a bounded domain Ω, provided the measures belong to Lp′(Ω) and g∈W1p-1,p′(∂Ω) (and 2NN+1<p).
In the case 1<p≤N we turn to the idea of renormalized solutions. In [1] a concept of renormalized solution was proposed for a Neumann problem in bounded domains and with nonnegative measures in L1 (see also [14]). However, to our knowledge, there is no proposed definition of renormalized solutions to Neumann problems such as (1.1) for general bounded Radon measures. In this work we propose such a definition and then prove existence of renormalized solutions for various types of nonlinearities.
Our approach to solving problem (1.1) is to turn it into an associated problem in the whole ℝN. Indeed, formally, if u is a solution to (1.1) then we expect that u¯, its even reflection across ∂ℝ+N, should be a solution to
(1.3)
-
Δ
p
u
¯
+
2
g
(
u
¯
)
ℋ
=
2
μ
in
ℝ
N
where ℋ is a normalized (N-1)-dimensional Hausdorff measure concentrated in ∂ℝ+N. Note however that not every solution of the above problem would yield a solution to (1.1), unless it is a symmetric solution, and so the problems are not equivalent.
The advantage of looking at this extended problem is that we can obtain a solution to (1.3) by applying the theory developed in [8, 23, 4, 16], to an increasing sequence of bounded domains. In order for this approach to work we need to establish some stability results. Then, to recover a solution to (1.1), we show that the solutions obtained through this process might in fact be taken to be symmetric with respect to ∂ℝ+N. It is worth mentioning that with our definition of renormalized solution to problem (1.1), u¯ becomes in fact a local renormalized solution of the associated problem in ℝN, as defined for example in [3] and [23].
This work is organized as follows. In Section 2 we collect all the relevant preliminary definitions and results we will need both to define renormalized solutions and to obtain the existence results. Since we consider measures and functions in both ℝN and ∂ℝ+N, we will need to consider the problem of obtaining well defined traces, as well as the interplay of the Bessel capacities defined in ℝN and ∂ℝ+N. In Section 3 we give the definitions of renormalized solutions in bounded domains and of local renormalized solutions in general domains, define renormalized solutions to problem (1.1), and state estimates and other results from [8] which will be used in the sequel.
In Section 4 we study the problem of obtaining local renormalized solutions to -Δpu=μ in ℝN. We prove the following.
Theorem 1.1.
Let μ¯∈Mb(RN) and 1<p≤N. Then there exists a local renormalized solution to
-
Δ
p
u
=
μ
¯
in
ℝ
N
.
The proof is based on two lemmas, which follow ideas introduced in [8] and [15]. Then, in Section 5, we show that the solution obtained by the above theorem is symmetric with respect to ∂ℝ+N. We do this by showing that, in a bounded domain, if a measure is concentrated in ∂ℝ+N and the domain is symmetric with respect to ∂ℝ+N, then any renormalized solution has the same symmetry. For this we use the partial uniqueness result obtained in [8]. This allows us to show the following existence result.
Theorem 1.2.
Let 1<p≤N and μ∈Mb(∂R+N). Then there exists a renormalized solution to
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
μ
on
∂
ℝ
+
N
.
In Section 6, we consider the problem of obtaining renormalized solutions to (1.1) in subcritical cases with absorption. The fact that g(s) is subcritical is expressed in the following assumption.
Assumption 1.3.
g
:
ℝ
→
ℝ
is a continuous function such that g(s)s≥0.
Define g~:ℝ+→ℝ by g~(s)=sup[-s,s]|g(t)|. If 1<p<N, we assume
∫
1
∞
g
~
(
s
)
s
-
p
(
N
-
2
)
+
1
N
-
p
𝑑
s
<
∞
.
If p=N, we assume that there exists γ>0 such that
∫
1
∞
g
~
(
s
)
e
-
γ
N
s
𝑑
s
<
∞
.
In the special case when g(s)=|s|q-1s, q≥0, Assumption 1.3 holds whenever
q
<
{
(
N
-
1
)
(
p
-
1
)
N
-
p
if
1
<
p
<
N
,
∞
if
p
=
N
.
Hence, we say that
q
c
:=
(
N
-
1
)
(
p
-
1
)
N
-
p
is a critical exponent for problem (1.1), and the problem is subcritical whenever q<qc.
Our approach is to use the existence results developed for problem (1.2) to obtain solutions to
(1.4)
{
-
Δ
p
u
+
g
(
u
)
ℋ
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
as an intermediate step towards solving (1.1). We obtain solutions to the above equation by solving (1.2) when g is multiplied by a sequence ζn(xN) that is concentrating at the origin and then letting n→∞. The main result in this regard is Lemma 6.6 where we show that if un are the solutions with nonlinear term ζng(un) then ζng(un) converges, in a suitable sense, to g(u)ℋ. We then prove the following.
Theorem 1.4.
Let μ∈Mb(∂R+N). Suppose g(s) satisfies Assumption 1.3. If p<N, then there exists a renormalized solution of (1.1). If p=N, then there exists a constant C(N) such that if ∥μ∥Mb≤C(N)γ1-N, then there exists a renormalized solution of (1.1).
In Section 7 we consider supercritical problems with absorption under the condition 1<p<N. Here we use mainly the work in [4]. As in the previous case, we obtain solutions to (1.4) as an intermediate step towards solving (1.1). The main tool for this is the improvement of an estimate of renormalized solutions, in terms of the Wolff potential Wα,s,NR[μ] of their respective measures, from a.e. in Ω to a.e. in any hyperplane. Our main existence result is Theorem 7.6. This theorem is then used to prove the following.
Theorem 1.5.
Assume 1<p<N and let g:R→R be a continuous nondecreasing odd function such that
|
g
(
s
)
|
≤
C
|
s
|
q
for all
|
s
|
≥
|
s
0
|
for some C>0, q>p-1, and s0∈R. If μ∈Mb(∂R+N) is absolutely continuous with respect to capp-1,qq-p+1,N-1, then there exists a renormalized solution to (1.1) with datum μ.
Finally, in Section 8 we consider nonnegative solutions to the problem with source
(1.5)
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
μ
+
u
q
on
∂
ℝ
+
N
,
where q>p-1, 1<p<N, and μ is nonnegative. Our work here follows closely the ideas in [16], particularly those used to treat problem (1.2) when Ω=ℝN. We begin by establishing necessary and sufficient conditions for existence of nonnegative renormalized solutions to (1.1). This leads to the following result.
Theorem 1.6.
Let 1<p<N, p-1<q , and assume μ in Mb(∂R+N) is nonnegative. Then the following are equivalent:
For some
ϵ
>
0
there exists a nonnegative renormalized solution to
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
ϵ
μ
+
u
q
on
∂
ℝ
+
N
,
satisfying
u
(
x
′
,
x
N
)
≤
C
(
p
,
q
,
N
,
ϵ
)
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
(
x
′
)
in
Ω
∩
ℝ
+
N
¯
,
where
Ω
=
Ω
1
∩
(
Ω
2
×
ℝ
)
, Ω2⊂ℝN-1, with cap1,p,N(Ω1c)=0 and |Ω2c|=0.
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
∈
L
q
(
∂
ℝ
+
N
)
and
W
1
-
1
/
p
,
p
,
N
-
1
[
(
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
)
q
]
≤
C
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
a.e. in
∂
ℝ
+
N
.
Moreover, if (2) holds with C small enough, then there exists a renormalized solution to (1.5).
As a consequence of this characterization we obtain a nonexistence theorem.
Theorem 1.7.
Let 1<p≤N, p-1<q, and μ∈Mb(∂R+N) nonnegative. If p=N, or p<N and q≤(N-1)(p-1)N-p, then there are no nontrivial nonnegative p-superharmonic solutions of -Δpu=2uqH+2μ in RN. In particular, there are no nontrivial nonnegative renormalized solutions of (1.5).
This result is a natural counterpart to the nonexistence result in [16]. It is also in agreement with the nonexistence result in [13] for the linear case (i.e., p=2) with q in the range [1,N-1N-2].
We finish with the problem of characterizing when a compact set K⊂∂ℝ+N is removable for (1.5) in the case μ≡0. We say that such a set K is removable if every nonnegative solution of
(1.6)
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
u
q
on
∂
ℝ
+
N
∖
K
,
can be extended to a solution of (1.5) with μ≡0. We show the following.
Theorem 1.8.
If 1<p<N and q>(N-1)(p-1)N-p, then a compact set K⊂∂R+N is removable for (1.6) if and only if capIp-1,q/(q-(p-1)),N-1(K)=0.
2 Preliminary Definitions and Results
For any measurable set E⊂ℝN we denote by |E| its Lebesgue measure. When E⊂∂ℝ+N≃ℝN-1, we take this measure to be the (N-1)-dimensional Lebesgue measure. We let BM(x) be the open ball of radius M>0 centered at x (simply BM when x=0). Depending on the context, when x∈∂ℝ+N this could be either an N-dimensional ball in ℝN or an (N-1)-dimensional ball in ∂ℝ+N. For any set E, let χE be its characteristic function. For any k>0 we let Tk(s)=min(k,max(-k,s)).
By an abuse of notation, we define
W
loc
1
,
p
(
ℝ
+
N
)
:=
⋂
M
∈
ℕ
W
1
,
p
(
B
M
∩
ℝ
+
N
)
.
Similarly, we define
L
loc
s
(
ℝ
+
N
)
:=
⋂
M
∈
ℕ
L
s
(
B
M
∩
ℝ
+
N
)
.
We remark that given a domain Ω, Ls(Ω) are the usual Lebesgue spaces, while W1,p(Ω) are the usual Sobolev spaces. The norm in the Ls(Ω) spaces will be written indistinctly as ∥⋅∥Ls(Ω), ∥⋅∥Ls, or simply ∥⋅∥s. The Ck(Ω) space, k∈ℕ∪{∞}, is the usual space of k-times continuously differentiable functions, and C0k(Ω) is the subspace of elements with compact support in Ω.
Bessel Capacities.
We recall the standard definition of Bessel capacities in ℝN (see [9] for details):
cap
α
,
p
,
N
(
E
)
=
inf
{
∥
f
∥
L
p
(
ℝ
N
)
p
:
f
∈
L
+
p
(
ℝ
N
)
for all
x
∈
ℝ
N
,
𝒢
α
∗
f
≥
1
for all
x
∈
E
}
,
where L+p(ℝN) is the subset of nonnegative elements of Lp(ℝN), and 𝒢α is the Bessel kernel of order α∈ℝ. If we use instead ℐα, the Riesz kernel of order α∈(0,N), we obtain the Riesz capacities, which we will denote by capIα,p,N(⋅).
We say that a property holds capα,p,N-quasi-everywhere in Ω (abbreviated as capα,p,N-q.e.) if there exists a set E such that the property holds in Ω∖E and capα,p,N(E)=0. Similarly, we say that a function ω is capα,p,N-quasi-continuous in Ω if for every ϵ>0 there is an open set E such that capα,p,N(E)<ϵ and ω∈C(Ω∖E). Unless otherwise stated, we assume that capα,p,N-quasi-continuous functions are capα,p,N-q.e. finite. Whenever we cannot assert that a capα,p,N-quasi-continuous function w is capα,p,N-q.e. finite, the statement ω∈C(Ω∖E) means that w:Ω∖E→[-∞,∞] is continuous with respect to the topology of the extended real line. Finally, we say that a set E⊂ℝN is quasi-open if for every ϵ>0 there exists an open set Ω such that E⊂Ω and cap1,p,N(Ω∖E)<ϵ. Clearly, countable unions of quasi-open sets are quasi-open. It is also immediate that if w is cap1,p,N-quasi-continuous, then the sets {w>k} and {w<k} are quasi-open.
We remark that, when dealing with a bounded domain Ω, it is more natural to define and use the so-called condenser capacity associated with Ω (see for example [8], [15], or [9, Section 7.6]). However, [12, Theorem 2.38] shows that the condenser capacity is equivalent to our definition of capacity whenever Ω=BM for any fixed M (see also [9, Section 2.7]). Since in our applications we always ultimately have Ω=BM for some M∈ℕ, we see that we can always assume the two definitions are equivalent.
Decomposition of Measures.
Let 𝔐b(ℝN) be the set of Radon measures of bounded total variation. For any Borel set Ω⊂ℝN we let 𝔐b(Ω) be the set of measures in 𝔐b(ℝN) supported in Ω. For measures μ∈𝔐b(Ω) we let ∥μ∥𝔐b=|μ|(Ω) be its total variation in Ω. Let us mention that we will work mainly with measures supported in ∂ℝ+N. Such measures can be naturally identified with measures in ℝN-1 and, whenever convenient, we will use this identification.
We let ℋ be the (N-1)-dimensional Hausdorff measure concentrated in ∂ℝ+N, normalized so that ℋ(E)=|E∩∂ℝ+N| for any measurable set E⊂ℝN. Then we define Ls(Ω∩∂ℝ+N):=Ls(Ω;dℋ) for any domain Ω and any 1≤s≤∞ (when Ω=ℝN we omit it from the notation). If a function g belongs to Lloc1(Ω∩∂ℝ+N), we write gℋ as shorthand for the measure gdℋ.
Any measure μ∈𝔐b(ℝN) can be uniquely decomposed as μ=μ0+μs, where μ0 is absolutely continuous with respect to capα,p,N, and μs is singular with respect to capα,p,N (see [11, Lemma 2.1]). Moreover, by the Jordan decomposition theorem, one can write uniquely μs=μs+-μs-, where μs+ and μs- are the positive and negative part of μs. In what follows we shall denote by 𝔐0(ℝN) the set of measures in 𝔐b(ℝN) that are absolutely continuous with respect to cap1,p,N. Similarly, 𝔐0(Ω) is the set of measures in 𝔐0(ℝN) which are supported in Ω.
The following result is proved in [5].
Theorem 2.1.
Let Ω be a bounded domain and μ∈Mb(Ω). Then μ∈M0(Ω) if and only if μ∈L1(Ω)+W-1,p′(Ω). Thus, if μ∈M0(Ω), then μ=f-divg in the sense of distributions for some functions f∈L1(Ω) and g∈(Lp′(Ω))N. Moreover, μ=f-divg also holds when acting on functions in W01,p(Ω)∩L∞(Ω).
We note that in the above result one can further assume that ∥f∥L1(Ω)+∥g∥W-1,p′(Ω) is bounded by 3∥μ∥𝔐b (see [4, Lemma 3.6]).
Lizorkin–Triebel Capacities.
Now we briefly consider the Lizorkin–Triebel spaces Fαp,q(ℝN). We refer the reader to [9] and [18] for definitions and details.
It can be shown that the spaces Fαp,q(ℝN) can be used to define corresponding Fαp,q(ℝN) capacities, which we denote by cap(⋅,Fαp,q(ℝN)) (see [9] for the details). The connection with the Bessel potential spaces is given by the fact that for any α>0, and 1<p<∞, there holds Fαp,2(ℝN)=Lα,p(ℝN) in the sense of normed spaces. As shown by [9, Proposition 4.4.4], for all α>0, 1<q<∞, and 1<p≤Nα, the Fαp,q(ℝN) capacity is equivalent to the corresponding Bessel Lα,p(ℝN) capacity. Moreover, it is proven in [18, Section 4.4] that any function w(x′,xN) in Fαp,q(ℝN) has a trace Tr(w)=w(x′,0) in Fα-1/pp,p(ℝN-1). Conversely, there exists an extension operator Ex:Fα-1/pp,p(ℝN-1)→Fαp,q(ℝN).
With the above results, one can show the following proposition, which shows that the “trace” of the cap1,p,N capacity in ∂ℝ+N is the cap1-1/p,p,N-1 capacity.
Proposition 2.2.
There exists a constant C(N,p) such that for all Borel sets E⊂RN and E′⊂∂R+N,
C
(
N
,
p
)
cap
1
,
p
,
N
(
E
)
≥
cap
1
-
1
/
p
,
p
,
N
-
1
(
E
∩
∂
ℝ
+
N
)
, and
C
(
N
,
p
)
cap
1
-
1
/
p
,
p
,
N
-
1
(
E
′
)
≥
cap
1
,
p
,
N
(
E
′
)
.
Thanks to the above result, we can describe the relationship between the decomposition of a measure in 𝔐b(ℝN-1) and its representative in 𝔐b(∂ℝ+N).
Proposition 2.3.
Let μ∈Mb(RN-1) and let μ=μ0+μs be its decomposition with respect to cap1-1/p,p,N-1. Let μ¯ denote its identification as an element of Mb(∂R+N). If μ¯=μ¯0+μ¯s is the decomposition of μ¯ with respect to cap1,p,N, then μ0¯=μ¯0 and μs¯=μ¯s. In particular, μs±¯=μ¯s±.
The proof is elementary and so we omit it.
Finer Properties of W1,p Functions.
Because F1p,2=L1,p=W1,p, every function w(x′,xN)∈W1,p(ℝN) has a trace w(x′,0) in F1-1/pp,p(ℝN-1). We now extend the notion of traces and study their regularity.
Proposition 2.4.
Let ω∈W1,p(R+N). Then ω has a cap1,p,N-quasi-continuous representative, defined in R+N¯, which is unique up to sets of zero cap1,p,N capacity. In particular, identifying ω with this representative, the trace of ω is cap1-1/p,p,N-1-quasi-continuous and unique cap1-1/p,p,N-1-q.e. in ∂R+N.
Proof.
Since ℝ+N is an extension domain, we consider ω as an element in W1,p(ℝN) and obtain the existence of a cap1,p,N-quasi-continuous representative which is unique in ℝ+N¯ modulo sets of zero capacity (see [9, Theorem 6.1.4]). In view of Proposition 2.2 the rest of the proposition follows easily. ∎
From now on we identify function in W1,p(ℝ+N) with their cap1,p,N-quasi-continuous representative in ℝ+N¯ and refer to their cap1-1/p,p,N-1-quasi-continuous trace in ∂ℝ+N whenever necessary. Note that this result also applies to functions in W1,p(ℝN), or in W01,p(BM) by identifying elements in this space with their extension by zero.
For a function ω∈Wloc1,p(ℝ+N) one can still define the boundary values of ω. Indeed, for any fixed m we have ω∈W1,p(Bm∩ℝ+N) and thus we can extend ω to a function in W1,p(ℝN) first by even reflection and then using that Bm is an extension domain. The resulting extension has a cap1,p,N-quasi-continuous representative, which we call ωm, coinciding with ω a.e. in Bm∩ℝ+N. If we take m′>m, then any cap1,p,N-quasi-continuous representative ωm′ coincides with ωm a.e. in Bm∩ℝ+N and thus, by [9, Theorem 6.1.4], ωm′=ωmcap1,p,N-q.e. in Bm∩ℝ+N¯. Hence, from now on, we identify functions in Wloc1,p(ℝ+N) with this locally defined, and cap1,p,N-q.e. unique, cap1,p,N-quasi-continuous representative in ℝ+N¯. In particular, if x′∈ℝN-1, we define the trace ω(x′,0) to be the value at (x′,0) of any representative ωm such that |x′|<m. By the above considerations, and Proposition 2.2, the trace is cap1-1/p,p,N-1-quasi-continuous and unique cap1-1/p,p,N-1-q.e. in ∂ℝ+N.
We shall make use of the following.
Proposition 2.5.
Let μ∈M0(∂R+N) and let w∈W1,p(RN). Then w is measurable with respect to μ. Furthermore, if the trace of w belongs to L∞(∂R+N) then it belongs to L∞(RN;dμ).
Proof.
By Proposition 2.4,w has a cap1-1/p,p,N-1-quasi-continuous trace in ∂ℝ+N. Since every cap1,p,N-quasi-continuous function coincides cap1,p,N-q.e. with a Borel function, it follows that w is measurable with respect to any measure μ∈𝔐0(∂ℝ+N). If moreover |w|≤k a.e. on ∂ℝ+N then, by an application of [9, Theorem 6.1.4], it holds |w|≤kcap1-1/p,p,N-1-q.e. on ∂ℝ+N. Since μ is absolutely continuous with respect to cap1-1/p,p,N-1, we see that |w|≤k μ-a.e. (see Proposition 2.3). ∎
We will use repeatedly the following fact: if u≤v a.e. in ℝN, where u and v are capα,p,N-quasi-continuous functions, then u≤vcapα,p,N-q.e. in ℝN (see [9, Theorem 6.1.4]).
p-Superharmonic Functions.
Let Ω be any domain. A p-superharmonic function is a lower semicontinuous function u:Ω→(-∞,∞], not identically infinite, such that for all open sets Ω′⊂⊂Ω and for all hp-harmonic in Ω′ and continuous in Ω′¯ we have that h≤u on ∂Ω′ implies h≤u in Ω′ (see [12]).
It is well known that if u is p-superharmonic, then its truncation min{u,k} belongs to Wloc1,p(Ω). This allows us to define its gradient in the same generalized sense as we will do for renormalized solutions (see Section 3), and in particular, it makes sense to define -Δpu in the sense of distributions. In particular, when we say that a p-superharmonic function u solves -Δpu=μ in Ω for some (not necessarily bounded) Radon measure μ, we mean it precisely in the sense of distributions, where the derivative of u is to be understood in the generalized sense. It is also known that if u is p-superharmonic function in Ω, then there exists a nonnegative Radon measure μ such that -Δpu=μ in 𝒟′(Ω).
3 Renormalized Solutions
Let Tk(s) be truncation by k, i.e., Tk(s)=min(k,max(-k,s)). Then for any measurable and a.e. finite u such that Tk(u)∈W01,p(Ω) (resp., Tk(u)∈Wloc1,p(ℝ+N)) for every k>0 there exists a measurable vector-valued function v:Ω→ℝN (resp., v:ℝ+N→ℝN) such that
∇
T
k
(
u
)
=
v
χ
{
|
u
|
<
k
}
a.e. in Ω (resp., a.e. in ℝ+N) for all k>0 (see [2, Lemma 2.1]). This function is unique a.e. and so we define v as the gradient of u and write ∇u=v. We note that, in general, v is not the gradient of u used in the definition of Sobolev spaces (see [8] for details).
Definition 3.1.
Let Ω be a bounded domain in ℝN. Let μ∈𝔐b(Ω) have a decomposition μ=μ0+μs with respect to cap1,p,N. Then a function u is a renormalized solution of
(3.1)
{
-
Δ
p
u
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
if
u is measurable, finite a.e., and Tk(u)∈W01,p(Ω) for all k>0,
|
∇
u
|
p
-
1
∈
L
q
(
Ω
)
for all 1≤q<NN-1,
there holds
∫
Ω
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
d
x
=
∫
Ω
w
𝑑
μ
0
+
∫
Ω
w
+
∞
𝑑
μ
s
+
-
∫
Ω
w
-
∞
𝑑
μ
s
-
for all w∈W01,p(Ω)∩L∞(Ω) satisfying the following condition: there exist k>0, r>N, and functions w±∞∈W1,r(Ω)∩L∞(Ω) such that w=w+∞ a.e. in {x∈Ω:u>k} and w=w-∞ a.e. in {x∈Ω:u<-k}.
By [8, Theorem 2.33] the last condition above can be replaced with several other equivalent ones. It is a fact that any renormalized solution has a cap1,p,N-quasi-continuous representative which is finite cap1,p,N-q.e. in Ω (see [8, Remark 2.18]). We always identify renormalized solutions with this representative.
The following theorem is proved in [8] using [2, Lemmas 4.1 and 4.2]. We note explicitly that the constants involved do not depend on the domain Ω.
Theorem 3.2.
Let u be a renormalized solution of (3.1). Then
(3.2)
∫
{
n
≤
|
u
|
<
n
+
k
}
|
∇
u
|
p
𝑑
x
≤
k
|
μ
|
(
Ω
)
for all
n
≥
0
and
k
>
0
.
If p<N, then for every k>0,
(3.3)
|
{
|
u
|
>
k
}
|
≤
C
(
N
,
p
)
(
|
μ
|
(
Ω
)
)
N
N
-
p
k
-
N
(
p
-
1
)
N
-
p
,
(3.4)
|
{
|
∇
u
|
>
k
}
|
≤
C
(
N
,
p
)
(
|
μ
|
(
Ω
)
)
N
N
-
1
k
-
N
(
p
-
1
)
N
-
1
.
If p=N, then for every k>0,
(3.5)
|
{
|
u
|
>
k
}
|
≤
C
(
r
,
N
,
p
)
(
|
μ
|
(
Ω
)
)
r
k
-
r
(
p
-
1
)
for every r>1, and
(3.6)
|
{
|
∇
u
|
>
k
}
|
≤
C
(
s
,
N
,
p
)
(
|
μ
|
(
Ω
)
)
N
N
-
1
k
-
s
for every s<N.
The following result is proven in [8, Section 5.1].
Theorem 3.3.
Let un be renormalized solutions to problem (3.1) with respective measures μn∈Mb(Ω). Assume ∥μn∥Mb are uniformly bounded. Then there exists a function u such that, up to a subsequence, un→u a.e. in Ω. Moreover, u satisfies (1) and (2) of the definition of renormalized solution, as well as all the estimates stated in Theorem 3.2 (with sup∥μn∥Mb instead of ∥μ∥Mb), and
∇
T
k
(
u
n
)
→
∇
T
k
(
u
)
and
∇
u
n
→
∇
u
a.e. in Ω,
|
∇
u
n
|
p
-
2
∇
u
n
→
|
∇
u
|
p
-
2
∇
u
strongly in
(
L
q
(
Ω
)
)
N
for any
1
≤
q
<
N
N
-
1
,
T
k
(
u
n
)
→
T
k
(
u
)
weakly in
W
0
1
,
p
(
Ω
)
.
It follows from [8, Remark 2.11] that the function u in the above theorem has a cap1,p,N-quasi-continuous representative which is finite cap1,p,N-q.e. in Ω. We identify u with this representative.
A closely related concept is the one of local renormalized solutions (see [3, 23]) on domains which are not necessarily bounded. We remark that the derivative here is to be understood in the same generalized sense as described previously.
Definition 3.4.
Let Ω be any domain in ℝN. Let μ∈𝔐b(Ω) have a decomposition μ=μ0+μs with respect to cap1,p,N. Then a function u is a local renormalized solution of
-
Δ
p
u
=
μ
in
Ω
if
u is measurable, finite a.e., and Tk(u)∈Wloc1,p(Ω) for all k>0,
|
∇
u
|
p
-
1
∈
L
loc
q
(
Ω
)
for all 1≤q<NN-1,
|
u
|
p
-
1
∈
L
loc
q
(
Ω
)
for all 1<q<NN-p (1<q<∞ if p=N),
there holds
∫
Ω
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
d
x
=
∫
Ω
w
𝑑
μ
0
+
∫
Ω
w
+
∞
𝑑
μ
s
+
-
∫
Ω
w
-
∞
𝑑
μ
s
-
for all w∈W1,p(Ω)∩L∞(Ω) compactly supported in Ω satisfying the following condition: there exist k>0, r>N, and functions w±∞∈W1,r(Ω)∩L∞(Ω) such that w=w+∞ a.e. in {x∈Ω:u>k} and w=w-∞ a.e. in {x∈Ω:u<-k}.
As in Definition 3.1, condition (4) can be replaced by some other equivalent conditions (see [3, Theorem 2.2]). We note that we have imposed that μ is bounded. This condition is not necessary for the definition, but we will use it when solving problem (1.1).
A fact that we will use frequently is that if μ is nonnegative and u is a local renormalized solution of -Δpu=μ in Ω, then u coincides a.e. with a p-superharmonic function solving the same equation (see [23, Theorem 4.3.2]).
Remark 3.5.
If Ω is bounded, then any renormalized solution of (3.1) is also a local renormalized solution of the corresponding equation. Indeed, we only need to show (3), but this follows from the identity
∫
Ω
|
u
|
α
d
x
=
∫
Ω
∫
0
∞
α
t
α
-
1
χ
(
t
)
[
0
,
|
u
|
]
d
t
d
x
=
∫
0
∞
α
t
α
-
1
|
{
|
u
|
≥
t
}
|
d
t
,
which holds for any measurable function u, and any α>0. From this identity one obtains the estimate
∫
Ω
|
u
|
α
d
x
≤
t
0
α
|
Ω
|
+
α
∫
t
0
∞
t
α
-
1
|
{
|
u
|
≥
t
}
|
d
t
.
In particular, if u is a renormalized solution in a bounded domain Ω, and p<N, then combining the above estimate and (3.3) we have u∈Ls(Ω) for 1<s<N(p-1)N-p. If p=N, then we use instead estimate (3.5).
We now define a renormalized solution to (1.1).
Definition 3.6.
Let μ∈𝔐b(∂ℝ+N) have a decomposition μ=μ0+μs with respect to cap1,p,N, and let g:ℝ→ℝ. A function u defined in ℝ+N is a renormalized solution to (1.1) if
u is measurable, finite a.e., and Tk(u)∈Wloc1,p(ℝ+N) for all k>0,
|
∇
u
|
p
-
1
∈
L
loc
q
(
ℝ
+
N
)
for all 1≤q<NN-1,
|
u
|
p
-
1
∈
L
loc
q
(
ℝ
+
N
)
for all 1<q<NN-p (1<q<∞ if p=N),
u is finite a.e. in ∂ℝ+N, and g(u)∈L1(∂ℝ+N),
there holds
∫
ℝ
+
N
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
d
x
+
∫
∂
ℝ
+
N
g
(
u
)
w
𝑑
x
′
=
∫
∂
ℝ
+
N
w
𝑑
μ
0
+
∫
∂
ℝ
+
N
w
+
∞
𝑑
μ
s
+
-
∫
∂
ℝ
+
N
w
-
∞
𝑑
μ
s
-
for all w∈W1,p(ℝ+N) compactly supported in ℝ+N¯, with trace in L∞(∂ℝ+N), and satisfying the following condition: there exist k>0, r>N, and functions w±∞∈W1,r(ℝ+N) such that w=w+∞ a.e. in {x∈ℝ+N:u>k} and w=w-∞ a.e. in {x∈ℝ+N:u<-k}.
It makes sense to talk about the boundary values of a renormalized solution since, in fact, any a.e. finite and measurable function u defined in ℝ+N such that Tk(u)∈Wloc1,p(ℝ+N) for all k>0 has a locally defined cap1,p,N-quasi-continuous representative in ℝ+N¯ which, however, could be infinite on a set of positive cap1,p,N capacity. Indeed, we know that we can locally identify Tk(u) with a cap1,p,N-quasi-continuous representative in ℝ+N¯. Then, it can be directly verified that v=supk∈ℕTk(u) defines (locally) a cap1,p,N-quasi-continuous function that coincides with u a.e. in ℝ+N and which is unique cap1,p,N-q.e. in ℝ+N¯. We note that similar considerations hold for a.e. finite and measurable functions u defined in ℝN such that Tk(u)∈Wloc1,p(ℝN).
From now on, we always identify renormalized solutions to (1.1) with their cap1,p,N-quasi-continuous representative in ℝ+N¯. In particular, under this identification, the trace of u is cap1-1/p,p,N-1-quasi-continuous and unique cap1-1/p,p,N-1-q.e. in ∂ℝ+N. Since u could be infinite on a set of positive capacity, we explicitly ask that the trace must be finite a.e. in ∂ℝ+N. We will show below that in fact renormalized solutions are always finite cap1,p,N-q.e. in ℝ+N¯.
If we consider cap1,p,N-quasi-continuous representatives in ℝ+N¯, then the condition w=w+∞ a.e. in {x∈ℝ+N:u>k} implies that w=w+∞ a.e. in {x∈∂ℝ+N:u>k}. To see this, apply [9, Theorem 6.1.4] to extend (w-w+∞)(u-k)+=0 from a.e. in ℝ+N¯ to cap1,p,N-q.e. in ℝ+N¯. Similarly, w=w-∞ a.e. in {x∈∂ℝ+N:u<-k}.
It is easy to verify that under the given assumptions all the integrals above are well defined and finite (see [8, Remark 2.14]). We just note that Proposition 2.4 guarantees that w has a well-defined trace, while Proposition 2.5 and the boundedness of μ0 gives w∈L1(∂ℝ+N;dμ0). Also, it follows directly from the definitions that if u is a renormalized solution of (1.1), then u¯, the extension of u by even reflection across ∂ℝ+N, is a local renormalized solution of -Δpu¯=μ~:=2μ-2g(u)ℋ in ℝN.
In the definition of renormalized solution we assumed u is finite a.e. in ∂ℝ+N. We now show that this is not a restriction, since g(u)∈L1(∂ℝ+N) implies that u must be finite a.e. in ∂ℝ+N. Indeed, by our definition of trace, and in view of Proposition 2.2, it will be enough to show that local renormalized solutions of -Δpu=μ in ℝN are finite cap1,p,N-q.e. in ℝN. We will obtain this as a consequence of the following local version of the estimates on level sets stated in Theorem 3.2.
Theorem 3.7.
Let u be a local renormalized solution of -Δpu=μ in Ω, and let Ω′ be such that Ω′⊂⊂Ω. Then
(3.7)
∫
{
|
u
|
<
k
}
∩
Ω
′
|
∇
u
|
p
𝑑
x
≤
C
(
p
,
Ω
,
Ω
′
,
μ
,
u
)
k
for all
k
>
0
.
Moreover, there exists k0(u,Ω,Ω′,p) such that for every k>k0 estimates (3.3)–(3.6) hold in Ω′, but where the constants on the right-hand side may depend also in Ω′ and u.
Proof.
Choose ϕ∈C0∞(Ω) such that 0≤ϕ≤1, ϕ≡1 in Ω′, and supp(ϕ)⊂Ω0⊂⊂Ω for some Ω0. Then, testing against ϕTk(u), we estimate
∫
Ω
′
|
∇
T
k
(
u
)
|
p
𝑑
x
≤
k
∥
∇
u
∥
L
p
-
1
(
Ω
0
)
∥
∇
ϕ
∥
∞
+
k
∥
μ
∥
𝔐
b
=
C
(
Ω
0
,
p
,
μ
,
u
)
k
which gives (3.7). Since u∈Ls(Ω0) for some s>0, by Chebyshev’s inequality we can choose k0 such that |{|u|>k2}∩Ω0|≤|Ω′|4 for all k≥k0. We let ck=(Tk(u))Ω′ be the average of Tk(u) in Ω′. Then, decomposing Ω′ into Ω′∩{|u|≤k2} and Ω′∩{|u|>k2}, we estimate |ck|≤3k4 for all k≥k0. Hence, if p<N, by Poincaré–Wirtinger inequality, Sobolev inequality, and (3.7), we obtain
∥
T
k
(
u
)
-
c
k
∥
L
q
(
Ω
′
)
≤
C
(
N
,
p
,
Ω
0
,
μ
,
u
)
k
1
p
,
where q=NpN-p. Since for all k≥k0 we have {|u|≥k}⊂{|Tk(u)-ck|≥k4}, we deduce
|
{
|
u
|
≥
k
}
∩
Ω
′
|
≤
(
4
∥
T
k
(
u
)
-
c
k
∥
L
q
(
B
M
)
k
)
q
≤
C
(
N
,
p
,
Ω
0
,
μ
,
u
)
k
q
(
1
-
p
p
)
,
which is the local version of (3.3). In the case p=N, the same procedure gives the local version of (3.5). The remaining estimates follow from the above ones just as in the proof of Theorem 3.2 in [8], using the results in [2]. ∎
Proposition 3.8.
Let u be a local renormalized solution of -Δpu=μ in RN. Then u is finite cap1,p,N-q.e. in RN. In particular, if v is a renormalized solution of (1.1) in the sense of definition 3.6, then the trace of v is finite cap1-1/p,p,N-1-q.e. in ∂R+N.
Proof.
It is enough to show that u is finite cap1,p,N-q.e. in ℝN. Fix M∈ℕ. By the previous theorem, with Ω=ℝN and Ω′=BM(0)=:BM, we can find k0(u,M,p) such that for all k≥k0 there holds
|
{
|
u
|
≥
k
}
∩
B
M
|
≤
|
B
M
|
4
.
Then c2k,M, the average of T2k(u) in BM, is bounded by 3k2 provided k≥k0.
Now, the function
ϕ
=
T
2
k
(
u
)
-
c
2
k
,
M
2
k
-
c
2
k
,
M
belongs to W1,p(BM), and by combining the Poincaré–Wirtinger inequality, estimate (3.7), and the above estimate, we conclude
∥
ϕ
∥
W
1
,
p
(
B
M
)
≤
C
(
p
,
N
,
M
,
u
)
|
2
k
-
c
2
k
,
M
|
∥
∇
T
2
k
(
u
)
∥
L
p
(
B
M
)
≤
C
(
p
,
N
,
M
,
μ
,
u
)
k
1
p
-
1
for any k≥k0. Further, we have ϕ=1 on the set {u≥2k}∩BM. Hence, by the definition of cap1,p,N we obtain
cap
1
,
p
,
N
(
{
u
≥
2
k
}
∩
B
M
)
≤
∥
ϕ
∥
W
1
,
p
(
B
M
)
p
≤
C
k
1
-
p
for any k≥k0. Since p>1, we conclude that cap1,p,N({u=+∞}∩BM)=0. In a similar way we can control the set where u=-∞. Since M∈ℕ is arbitrary, this concludes the proof. ∎
4 Local Renormalized Solutions in ℝN
The proof of Theorem 1.1 rests in the following two lemmas. Whenever convenient, we extend functions by zero.
Lemma 4.1.
Let 1<p≤N. Let νm∈Mb(RN) be a sequence of measures such that |νm|(Bm)≤C1<∞ for all m∈N. Let um be renormalized solutions to
{
-
Δ
p
u
m
=
ν
m
in
B
m
,
u
m
=
0
on
∂
B
m
.
Then there exists a function u such that, up to a subsequence, um→u a.e. in RN. Moreover:
u
is measurable and finite
cap
1
,
p
,
N
-
q.e.,
T
k
(
u
m
)
→
T
k
(
u
)
weakly in
W
1
,
p
(
B
M
)
for any fixed
k
>
0
and
M
∈
ℕ
, and
∇
T
k
(
u
m
)
→
∇
T
k
(
u
)
a.e. in
ℝ
N
for any
k
>
0
,
∇
u
m
→
∇
u
a.e. and
|
∇
u
m
|
p
-
2
∇
u
m
→
|
∇
u
|
p
-
2
∇
u
strongly in
(
L
q
(
B
M
)
)
N
for any
M
∈
ℕ
and
1
≤
q
<
N
N
-
1
,
|
u
|
p
-
1
∈
L
loc
q
(
ℝ
N
)
for all
1
<
q
<
N
N
-
p
(
1
<
q
<
∞
if
p
=
N
).
Proof.
To begin we note that each um satisfies the estimates stated in Theorem 3.2 uniformly in the sense that they hold with |νm|(Bm) replaced by C1. Now fix any M∈ℕ, k∈ℕ, and σ>0. Observe that the set {x∈BM:|um-un|>σ} is contained in
{
x
∈
B
M
:
|
u
m
|
>
k
}
∪
{
x
∈
B
M
:
|
u
n
|
>
k
}
∪
{
x
∈
B
M
:
|
T
k
(
u
m
)
-
T
k
(
u
n
)
|
>
σ
}
.
Thanks to (3.3) and (3.5) the measure of the first two sets is arbitrarily small, independent of m and n, provided k is large enough.
For each fixed k estimate (3.2) gives that the sequence {Tk(um)}m is uniformly bounded in W1,p(BM) for any fixed k and M. Since the injection W1,p(BM)↪Lp(BM) is compact, this means that {Tk(um)}m has a subsequence that converges strongly in Lp(BM), and hence, that it is a Cauchy subsequence in measure in BM.
Now fix M and apply a diagonalization argument to obtain a diagonal sequence {um,m}m, which we relabel as {um}m, which is a Cauchy sequence in measure. Hence, passing to a subsequence, there exists a measurable and a.e. finite function vM such that um→vM a.e. in BM. Proceeding in a similar way but now with respect to M∈ℕ, we can obtain a subsequence {um,m}m⊂{um}m such that for every M∈ℕ, um→vM a.e. in BM. Relabeling this subsequence as {um}m, we see that there exists a measurable and a.e. finite function u such that um→u a.e. in ℝN and u=vM a.e. in BM.
Almost all the properties listed in the lemma can be obtained as in [8, Section 5], so we omit the details (see also [23, proof of Theorem 4.3.8]). We only have to show that u is finite cap1,p,N-q.e. in ℝN.
Fix M∈ℕ and let ck,m,M and ck,M be the averages of Tk(um) and Tk(u), respectively, in BM. By estimate (3.3) and (3.5) we can choose k0>0 such that for all k≥k0 and for all m∈ℕ we have |{|um|≥k}|≤|BM|4. Thus, |c2k,m,M|≤3k2 for any k≥k0. By Lebesgue’s Dominated Convergence Theorem we have ck,M=limm→∞ck,m,M and so we also get |c2k,M|≤3k2 for any k≥k0. Now, to finish, we can proceed as in the proof of Proposition 3.8, by considering the function ϕ=T2k(u)-c2k,M2k-c2k,M. ∎
Note that (1) in Lemma 4.1 implies that u has a cap1,p,N-quasi-continuous representative, which we identify with u.
Lemma 4.2.
Let μ¯∈Mb(RN) and assume gm and g are measurable functions defined in ∂R+N such that ∥gm∥L1(Bm∩∂R+N)+∥g∥L1(∂R+N)≤C1<∞ for some positive constant C1. Let um be renormalized solutions to
{
-
Δ
p
u
m
=
μ
¯
m
-
g
m
ℋ
in
B
m
,
u
m
=
0
on
∂
B
m
,
where μ¯m is the restriction of μ¯ to Bm. Assume um→u a.e. in RN, where u is a function satisfying properties (1), (2), and (3) in Lemma 4.1. Suppose also that
(4.1)
lim
m
→
∞
∫
B
M
∩
∂
ℝ
+
N
ϕ
m
g
m
𝑑
x
′
=
∫
B
M
∩
∂
ℝ
+
N
ϕ
g
𝑑
x
′
for any M∈N and any sequence {ϕm}m converging to ϕ both a.e. in BM and weakly in W01,p(BM) and such that ϕm is uniformly bounded in L∞(BM). Then u is a local renormalized solution of
-
Δ
p
u
=
μ
¯
-
g
ℋ
in
ℝ
N
.
Moreover, Tk(um)→Tk(u) strongly in W1,p(BM) for any fixed k>0 and M∈N.
Proof.
We show first that u solves the desired equation. Since we follow very closely the proof of Theorem 4.1 in [15], we only point out the main ideas.
By Theorem 2.1 we have (μ¯m)0=(μ¯0)m=fm-divhm in 𝒟′(Bm) for some fm∈L1(Bm) and hm∈(Lp′(Bm))N. Then, by [15, Lemma 3.1] there exists a set U⊂(0,∞) with Uc of measure zero such that for each um and k∈U there exists some measures αm,k± such that
∫
{
|
u
m
|
<
k
}
(
|
∇
T
k
(
u
m
)
|
p
-
2
∇
T
k
(
u
m
)
-
h
m
)
⋅
∇
v
d
x
=
∫
{
u
m
=
k
}
v
𝑑
α
m
,
k
+
-
∫
{
u
m
=
-
k
}
v
𝑑
α
m
,
k
-
+
∫
{
|
u
m
|
<
k
}
v
f
m
𝑑
x
-
∫
{
|
u
m
|
<
k
}
∩
∂
ℝ
+
N
v
g
m
𝑑
x
′
for every v∈W01,p(Bm)∩L∞(Bm).
Given M∈ℕ, let EM={k∈ℝ+:|{x∈BM:|u|=k}|>0} and write FM=(EM)c. Note that χ{|um|<k}→χ{|u|<k} a.e. in BM and weakly-∗ in L∞(BM) for all k∈FM. Putting ϕ∈C0∞(BM) as test function in the above identity and fixing k∈FM, we may use the hypotheses to pass to the limit as m→∞ in both the left-hand side and the third integral on the right-hand side.
Note that the functions gm,k=gmχ{|um|<k} are uniformly bounded in L1(BM∩∂ℝ+N). Then, passing to a subsequence depending on k, there exist a measure τk∈𝔐b(ℝN) which is the weak-∗ limit of the measures gm,k. Similarly, just as in the proof of Theorem 4.1 in [15], since μ¯m and gm and are uniformly bounded as measures, for every k in some subset V, with Vc of measure zero, there exist measures λk± which are the weak-∗ limit of (a subsequence of) αm,k±. In particular, we see that for any ϕ∈C0∞(BM) and k∈KM=FM∩U∩V,
(4.2)
∫
{
|
u
|
<
k
}
∩
B
M
(
|
∇
T
k
(
u
)
|
p
-
2
∇
T
k
(
u
)
-
h
M
)
⋅
∇
ϕ
d
x
=
∫
{
|
u
|
<
k
}
∩
B
M
ϕ
f
M
𝑑
x
+
∫
B
M
ϕ
d
(
-
τ
k
+
λ
k
+
-
λ
k
-
)
.
By Theorem 2.1, one concludes from the above that .-τk+λk+-λk-|BM belongs to 𝔐0(BM) and that (4.2) extends to ϕ∈W01,p(BM)∩L∞(BM).
The sets {x∈BM:|u|>k} and {x∈BM:|u|<k} are quasi-open. Then, as in [15], we conclude from (4.2) that
.
(
-
τ
k
+
λ
k
+
-
λ
k
-
)
|
{
|
u
|
>
k
}
∩
B
M
=
0
for any k in KM, and that there exists a measure ν0∈𝔐0(ℝN) with support in BM such that
.
ν
0
|
{
|
u
|
<
k
}
=
.
(
-
τ
h
+
λ
h
+
-
λ
h
-
)
|
{
|
u
|
<
k
}
∩
B
M
for any h≥k in KM. We also define
ν
k
+
=
.
(
-
τ
k
+
λ
k
+
-
λ
k
-
)
|
{
u
=
k
}
∩
B
M
,
ν
k
-
=
-
.
(
-
τ
k
+
λ
k
+
-
λ
k
-
)
|
{
u
=
-
k
}
∩
B
M
.
Following the ideas in [15], and using (4.1), it is not hard to show that there exists a sequence of positive numbers k∈KM going to infinity such that νk±→ν± weakly-∗ in 𝔐b(BM) as k→∞, for some nonnegative measures ν±. Moreover, one can show that μ¯s-gℋ=ν0+ν+-ν- in BM, ν+≤μ¯s+, and ν-≤μ¯s-. This implies in particular that ν+ and ν- are singular with respect to cap1,p,N, and since μ¯s-gℋ=ν0+ν+-ν- we conclude that ν0≡-gℋ. Recalling that μ¯s+ and μ¯s- have disjoint support we further conclude ν+=μ¯s+ and ν-=μ¯s- in BM. In particular, this allows us to rewrite (4.2) as
(4.3)
∫
{
|
u
|
<
k
}
∩
B
M
(
|
∇
T
k
(
u
)
|
p
-
2
∇
T
k
(
u
)
-
h
M
)
⋅
∇
ϕ
d
x
=
∫
{
|
u
|
<
k
}
∩
B
M
ϕ
f
M
𝑑
x
-
∫
{
|
u
|
<
k
}
∩
B
M
ϕ
g
𝑑
x
′
+
∫
{
u
=
k
}
∩
B
M
ϕ
𝑑
ν
k
+
-
∫
{
u
=
-
k
}
∩
B
M
ϕ
𝑑
ν
k
-
for any ϕ∈W01,p(BM)∩L∞(BM) and k∈KM.
Now let w∈W1,∞(ℝ) with w′ compactly supported, and let ϕ∈W1,r(ℝN), for some r>N, be compactly supported and such that w(u)ϕ∈W1,p(ℝN). We write w(±∞)=lims→±∞w(s). Choosing M∈ℕ large enough, we can assume w(u)ϕ∈W01,p(BM) so that w(u)ϕ is a valid test function for (4.3) with kj∈KM chosen to be the sequence such that νkj±→ν± weakly-∗ as kj→∞. Its not hard to show that we can take kj→∞ and obtain
∫
B
M
|
∇
u
|
p
-
2
∇
u
⋅
∇
(
w
(
u
)
ϕ
)
d
x
=
∫
B
M
w
(
u
)
ϕ
𝑑
μ
¯
0
-
∫
B
M
∩
∂
ℝ
+
N
w
(
u
)
ϕ
g
𝑑
x
′
+
w
(
+
∞
)
∫
B
M
ϕ
𝑑
μ
¯
s
+
-
w
(
-
∞
)
∫
B
M
ϕ
𝑑
μ
¯
s
-
.
Hence, by the results in [3], u is a local renormalized solution of -Δpu=μ¯-gℋ in ℝN.
Now we show the strong convergence of the truncates. Fix M∈ℕ, k>0, and ϕ∈C0∞(BM). Considering the identities obtained by testing against Tk(um)ϕ in the definition of um as renormalized solution, and against Tk(u)ϕ in the definition of u as renormalized solution, it is possible to show, using condition (4.1), that
lim
m
→
∞
∫
B
M
ϕ
|
∇
T
k
(
u
m
)
|
p
𝑑
x
=
∫
B
M
ϕ
|
∇
T
k
(
u
)
|
p
𝑑
x
.
This implies that, for any M>M′∈ℕ,
lim
m
→
∞
∥
∇
T
k
(
u
m
)
∥
L
p
(
B
M
′
)
=
∥
∇
T
k
(
u
)
∥
L
p
(
B
M
′
)
.
Using the above, the inequality ||a+b|-|a|-|b||≤2|b|, and the fact that ∇Tk(um)→∇Tk(u) a.e. in BM′, we obtain that |∇Tk(um)|p→|∇Tk(u)|p strongly in L1(BM′). Then, by Vitali’s Theorem, ∇Tk(um)→∇Tk(u) strongly in (Lp(BM′))N, from which the claim follows. ∎
Proof of Theorem 1.1..
Let μ¯m be the restriction of μ¯ to Bm. Since |μ¯m|(Bm)≤μ¯(ℝN)<∞, we can apply Lemma 4.1 and Lemma 4.2 with g=gm≡0. ∎
5 Symmetric Solutions
The following result was announced in the introduction.
Theorem 5.1.
Let Ω be any bounded domain in RN that is symmetric with respect to the hyperplane ∂R+N. Let μ¯∈Mb(Ω) be supported in ∂R+N∩Ω and let u be a renormalized solution to
{
-
Δ
p
u
=
μ
¯
in
Ω
,
u
=
0
on
∂
Ω
.
Then u(x′,xN)=u(x′,-xN) a.e. in Ω.
Proof.
In what follows we write Ω+=Ω∩ℝ+N and for any f defined in Ω we denote by f∗ its reflection with respect to ∂ℝ+N, i.e., f∗(x′,xN)=f(x′,-xN).
It can be show that the function u∗ is also a renormalized solution of the above problem and that Tk(u)-Tk(u∗)∈W01,p(Ω+) for any k>0. By the equivalence of definitions of renormalized solutions, we have that for every k>0 there exist two nonnegative measures λk+, λk-∈𝔐0(Ω) supported in {u=k} and {u=-k}, respectively, such that λk±→μ¯s± as k→∞ in the narrow topology of measures (see [8]), and the truncations Tk(u) satisfy
(5.1)
∫
{
|
u
|
<
k
}
|
∇
T
k
(
u
)
|
p
-
2
∇
T
k
(
u
)
⋅
∇
v
d
x
=
∫
{
u
=
k
}
v
𝑑
λ
k
+
-
∫
{
u
=
-
k
}
v
𝑑
λ
k
-
+
∫
{
|
u
|
<
k
}
v
𝑑
μ
¯
0
for every v∈W01,p(Ω)∩L∞(Ω). Note that λk±(Ω+)→μ¯s±(Ω+)=0 as k→∞ since μ¯s± is supported in ∂ℝ+N. Arguing in the same way for u∗, we obtain sequences (λk±)∗ converging to μ¯s± such that a similar identity to (5.1) holds with u∗ in place of u. Now, if we extend Tk(u)-Tk(u∗) by 0 outside Ω+, then we obtain that Tk(u)-Tk(u∗)∈W01,p(Ω)∩L∞(Ω) is a valid test function for (5.1) which vanishes in ∂ℝ+N∩Ω. Hence, we can test against Tk(u)-Tk(u∗) in the equations satisfied by u∗ and u, and subtracting, we get
(5.2)
∫
Ω
+
[
|
∇
T
k
(
u
)
|
p
-
2
∇
T
k
(
u
)
-
|
∇
T
k
(
u
∗
)
|
p
-
2
∇
T
k
(
u
∗
)
]
⋅
∇
[
T
k
(
u
)
-
T
k
(
u
∗
)
]
d
x
=
∫
Ω
+
[
T
k
(
u
)
-
T
k
(
u
∗
)
]
𝑑
λ
k
+
-
∫
Ω
+
[
T
k
(
u
)
-
T
k
(
u
∗
)
]
𝑑
λ
k
-
-
∫
Ω
+
[
T
k
(
u
)
-
T
k
(
u
∗
)
]
d
(
λ
k
+
)
∗
+
∫
Ω
+
[
T
k
(
u
)
-
T
k
(
u
∗
)
]
d
(
λ
k
-
)
∗
.
Using the well-known inequality
(5.3)
∑
i
=
1
N
(
|
z
|
p
-
2
z
i
-
|
ζ
|
p
-
2
ζ
i
)
(
z
i
-
ζ
i
)
≥
C
{
|
z
-
ζ
|
p
,
p
≥
2
,
|
z
-
ζ
|
2
(
|
z
|
+
|
ζ
|
)
p
-
2
,
p
≤
2
,
it follows immediately from (5.2) that
∫
Ω
+
|
∇
T
k
(
u
)
-
∇
T
k
(
u
∗
)
|
p
𝑑
x
≤
C
(
p
)
k
[
|
λ
k
+
|
(
Ω
+
)
+
|
λ
k
-
|
(
Ω
+
)
+
|
(
λ
k
+
)
∗
|
(
Ω
+
)
+
|
(
λ
k
-
)
∗
|
(
Ω
+
)
]
when p≥2. When 1<p<2, we use Hölder’s inequality first to get
∫
Ω
+
|
∇
T
k
(
u
)
-
∇
T
k
(
u
∗
)
|
p
𝑑
x
≤
{
∫
Ω
+
|
∇
T
k
(
u
)
-
∇
T
k
(
u
∗
)
|
2
(
|
∇
T
k
(
u
)
|
+
|
∇
T
k
(
u
∗
)
|
)
2
-
p
𝑑
x
}
p
2
{
∫
Ω
+
(
|
∇
T
k
(
u
)
|
+
|
∇
T
k
(
u
∗
)
|
)
p
𝑑
x
}
2
-
p
2
,
which then by (5.3), (5.2), and (3.2) yields
∫
Ω
+
|
∇
T
k
(
u
)
-
∇
T
k
(
u
∗
)
|
p
𝑑
x
≤
C
k
[
|
λ
k
+
|
(
Ω
+
)
+
|
λ
k
-
|
(
Ω
+
)
+
|
(
λ
k
+
)
∗
|
(
Ω
+
)
+
|
(
λ
k
-
)
∗
|
(
Ω
+
)
]
p
2
{
|
μ
¯
|
(
Ω
)
}
2
-
p
2
.
Thus we see that for any 1<p≤N there holds
1
k
∫
Ω
+
|
∇
T
k
(
u
)
-
∇
T
k
(
u
∗
)
|
p
𝑑
x
→
0
as k→∞. By symmetry, the same is true in Ω∩ℝ-N. Hence we can apply the partial uniqueness result stated in [8, Theorem 10.4] to conclude that u=u∗ a.e. in Ω. ∎
The following theorem allows us to obtains solutions to (1.1) provided we have symmetry.
Theorem 5.2.
Let 1<p≤N and μ∈Mb(∂R+N). Suppose u is a local renormalized solution of -Δpu=2μ in RN that is symmetric with respect to the hyperplane ∂R+N. Then the restriction of u to R+N is a renormalized solution of
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
μ
on
∂
ℝ
+
N
.
Proof.
It is clear from the definition of local renormalized solution that the restriction of u to ℝ+N satisfies conditions (1), (2), and (3) of Definition 3.6. Hence, we only need to show that (5) holds.
Assume that w∈W1,p(ℝ+N) is a valid test function as indicated in (5). Choose L such that |w|≤L a.e. in ∂ℝ+N and |w±∞|≤L in ℝ+N. Let us extend w and w±∞ to ℝN by even reflection, i.e., w(x′,xN)=w(x′,-xN) for xN<0 and similarly for w±∞. Note that since u is symmetric with respect to ∂ℝ+N we have that the extension of w is a valid test function for the definition of u as renormalized solution of the problem in ℝN.
Next, we let
Φ
ϵ
=
{
0
,
x
N
≤
0
,
x
N
ϵ
,
0
<
x
N
≤
ϵ
,
1
,
ϵ
≤
x
N
,
Ψ
ϵ
=
{
0
,
x
N
≤
-
ϵ
,
x
N
+
ϵ
ϵ
,
-
ϵ
<
x
N
≤
0
,
1
,
0
≤
x
N
.
Then for any l>L we see that Tl(w)Ψϵ∈W1,p(ℝN)∩L∞(ℝN) is an adequate test function and thus we get
∫
ℝ
N
Ψ
ϵ
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
+
1
ϵ
∫
{
-
ϵ
≤
x
N
≤
0
}
T
l
(
w
)
|
∇
u
|
p
-
2
u
x
N
𝑑
x
=
2
{
∫
∂
ℝ
+
N
w
𝑑
μ
0
+
∫
∂
ℝ
+
N
w
+
∞
𝑑
μ
s
+
-
∫
∂
ℝ
+
N
w
-
∞
𝑑
μ
s
-
}
.
Taking now Tl(w)Φϵ as test function and using uxN(x′,xN)=-uxN(x′,-xN) a.e., we get
∫
ℝ
N
Φ
ϵ
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
-
1
ϵ
∫
{
-
ϵ
≤
x
N
≤
0
}
T
l
(
w
)
|
∇
u
|
p
-
2
u
x
N
𝑑
x
=
0
.
Adding up the previous equalities, we conclude
∫
ℝ
N
Ψ
ϵ
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
+
∫
ℝ
N
Φ
ϵ
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
=
2
{
∫
∂
ℝ
+
N
w
𝑑
μ
0
+
∫
∂
ℝ
+
N
w
+
∞
𝑑
μ
s
+
-
∫
∂
ℝ
+
N
w
-
∞
𝑑
μ
s
-
}
and by Lebesgue’s Dominated Convergence Theorem we let ϵ→0 to obtain
∫
ℝ
+
N
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
=
∫
∂
ℝ
+
N
w
𝑑
μ
0
+
∫
∂
ℝ
+
N
w
+
∞
𝑑
μ
s
+
-
∫
∂
ℝ
+
N
w
-
∞
𝑑
μ
s
-
.
Writing
∫
ℝ
+
N
|
∇
u
|
p
-
2
∇
u
⋅
∇
T
l
(
w
)
𝑑
x
=
∫
ℝ
+
N
∩
{
|
u
|
≤
k
}
|
∇
T
k
(
u
)
|
p
-
2
∇
T
k
(
u
)
⋅
∇
T
l
(
w
)
𝑑
x
+
∫
ℝ
+
N
∩
{
u
>
k
}
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
+
∞
d
x
+
∫
ℝ
+
N
∩
{
u
<
-
k
}
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
-
∞
d
x
,
we use the fact that ∇Tl(w)→∇w weakly in (Lp(ℝ+N))N to take l→∞ above, and conclude
∫
ℝ
+
N
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
d
x
=
∫
∂
ℝ
+
N
w
𝑑
μ
0
+
∫
∂
ℝ
+
N
w
+
∞
𝑑
μ
s
+
-
∫
∂
ℝ
+
N
w
-
∞
𝑑
μ
s
-
,
thus completing the proof of the theorem. ∎
Proof of Theorem 1.2.
Apply Theorem 1.1 to obtain a local renormalized solution to -Δpu=2μ in ℝN. By the construction of u, and in view of Theorem 5.1, u is symmetric with respect to ∂ℝ+N. Then the result follows from an application of the previous theorem. ∎
6 Nonlinear Problems with Absorption. The Subcritical Case
We let
(6.1)
ζ
(
t
)
=
1
π
(
1
1
+
t
2
)
,
ζ
n
(
t
)
=
n
ζ
(
n
t
)
,
g
n
(
x
,
s
)
=
ζ
n
(
x
N
)
g
(
s
)
.
Note that ζ∈C∞(ℝ) and ∥ζ∥L1=1. We need the following definitions.
Definition 6.1.
Let Ω be a bounded domain, μ∈𝔐b(Ω), and g:ℝ→ℝ. Then a function u defined in Ω is a renormalized solution to problem (1.4) if u is finite a.e. in Ω∩∂ℝ+N, g(u)∈L1(Ω∩∂ℝ+N) and u is a renormalized solution to problem (3.1) with datum μ-g(u)ℋ in the sense of Definition 3.1.
Similarly, if g:Ω×ℝ→ℝ, then a function u defined in Ω is a renormalized solution to problem (1.2) if g(x,u)∈L1(Ω) and u is a renormalized solution to problem (3.1) with datum μ-g(x,u) in the sense of Definition 3.1.
The following result is obtained in [23, proof of Theorem 5.1.2].
Proposition 6.2.
Let u be a renormalized solution to problem (1.2), where g(x,⋅) is continuous and satisfies g(x,s)s≥0 for all x∈Ω and s∈R. Then ∥g(x,u)∥1≤|μ|(Ω).
The next lemma collects some relationships between capacities and Lebesgue measure (see [9] and Proposition 2.2).
Lemma 6.3.
Let 1<p≤N. There exist positive constants C1(M,N,p), C2(M,N,p), and C3(N,p) such that for all Borel sets E⊂BM⊂RN the following hold:
C
1
|
E
∩
∂
ℝ
+
N
|
p
≤
cap
1
-
1
/
p
,
p
,
N
-
1
(
E
∩
∂
ℝ
+
N
)
≤
C
3
cap
1
,
p
,
N
(
E
)
,
|
E
|
+
|
E
∩
{
x
N
=
t
}
|
≤
C
2
cap
1
,
p
,
N
(
E
)
1
p
,
Now we obtain estimates similar to (3.3) and (3.5) but on hyperplanes.
Lemma 6.4.
Suppose f is cap1,p,N-quasi-continuous in RN and Tk(f)∈W01,p(BM) satisfies ∥∇Tk(f)∥pp≤kC1. If 1<p<N, then there exists a constant C(N,p,BM) such that for any t∈R,
|
{
x
∈
B
M
∩
{
x
N
=
t
}
:
|
f
|
>
k
}
|
≤
C
(
N
,
p
,
B
M
)
C
1
N
-
1
N
-
p
k
-
(
N
-
1
)
(
p
-
1
)
N
-
p
.
If p=N, then there exists constants C(N,BM) and c(N) such that for any t∈R,
|
{
x
∈
B
M
∩
{
x
N
=
t
}
:
|
f
|
>
k
}
|
≤
C
(
N
,
B
M
)
e
-
c
(
N
)
k
(
C
1
)
-
1
N
-
1
.
Proof.
Suppose 1<p<N. By Sobolev’s embedding in Lizorkin–Triebel spaces (see [19]), the trace inequality, and the Poincaré inequality, we have for q=p(N-1)N-p that
∥
T
k
(
f
)
∥
L
q
(
B
M
∩
{
x
N
=
t
}
)
≤
C
(
N
,
p
,
B
M
)
(
k
C
1
)
1
p
.
Since {|f|>k}={|Tk(f)|≥k}, we conclude
|
{
x
∈
B
M
∩
{
x
N
=
t
}
:
|
f
|
>
k
}
|
≤
(
∥
T
k
(
f
)
∥
q
k
)
q
≤
C
(
N
,
p
,
B
M
)
C
1
q
p
k
(
1
p
-
1
)
q
,
which finishes the proof for the case p<N. When p=N, the results in [6] show that
∫
B
M
∩
{
x
N
=
t
}
e
c
1
(
|
T
k
(
f
)
|
∥
∇
T
k
(
f
)
∥
L
N
(
B
M
)
)
N
N
-
1
𝑑
x
′
≤
c
2
(
N
,
B
M
)
for some constants c1(N) and c2(N,BM). Since ∥∇Tk(f)∥N≤(kC1)1N, we conclude
|
{
x
∈
B
M
∩
{
x
N
=
t
}
:
|
f
|
>
k
}
|
e
c
1
k
C
1
1
1
-
N
≤
c
2
,
which gives the desired bound. ∎
Remark 6.5.
Let us note that we can also show |{x∈BM:|f|>k}|≤C(N,BM)exp(-c(N)kC111-N) when p=N.
The next two lemmas will allow us to obtain solutions to (1.4) from solutions to (1.2).
Lemma 6.6.
Fix m>0. Suppose g satisfies part (1) of Assumption 1.3, let g~ be defined as in part (2) of Assumption 1.3, and let gn be defined by (6.1). Let un→u a.e. in Bm, where u,un are cap1,p,N-quasi-continuous in RN. Assume also that Tk(un)→Tk(u) weakly in W01,p(Bm) for any k>0. Define
δ
t
(
n
,
h
)
=
∫
B
m
∩
{
|
u
n
|
≥
h
}
∩
{
x
N
=
t
}
g
~
(
|
u
n
|
)
(
x
′
,
t
)
𝑑
x
′
,
δ
(
h
)
=
∫
B
m
∩
{
|
u
|
≥
h
}
∩
{
x
N
=
0
}
g
~
(
|
u
|
)
(
x
′
,
0
)
𝑑
x
′
.
If δt(n,h)→0 and δ(h)→0 as h→∞, uniformly in t and n, then
(6.2)
lim
n
→
∞
∫
B
m
ϕ
n
g
n
(
x
,
u
n
)
𝑑
x
=
∫
B
m
∩
∂
ℝ
+
N
ϕ
g
(
u
)
𝑑
x
′
whenever {ϕn}n is a bounded subset of L∞(Bm) such that ϕn→ϕ both a.e. in Bm and weakly in W01,p(Bm).
Proof.
For any n and k we write Enk={|un|<k} and Ek={|u|<k}. Note that
∫
B
m
|
ϕ
n
|
|
g
n
(
x
,
u
n
)
-
g
n
(
x
,
T
k
(
u
n
)
)
|
𝑑
x
≤
2
∥
ϕ
n
∥
∞
∫
(
E
n
k
)
c
ζ
n
(
x
N
)
g
~
(
|
u
n
|
)
𝑑
x
.
For all n we can estimate
∫
(
E
n
k
)
c
ζ
n
(
x
N
)
g
~
(
|
u
n
|
)
𝑑
x
≤
∫
ℝ
ζ
n
(
t
)
δ
t
(
n
,
k
)
𝑑
t
≤
∥
ζ
n
∥
1
∥
δ
t
(
n
,
k
)
∥
∞
,
and since δt(n,k)→0 uniformly, we conclude the above goes to zero as k→∞, uniformly in t and n. In a similar way we can write Γ=Bm∩∂ℝ+N and estimate
∫
Γ
|
ϕ
|
|
g
(
u
)
-
g
(
T
k
(
u
)
)
|
𝑑
x
′
=
∫
Γ
∩
(
E
k
)
c
|
ϕ
|
|
g
(
u
)
-
g
(
T
k
(
u
)
)
|
≤
2
∥
ϕ
∥
∞
δ
(
k
)
→
0
as k→∞. Collecting the above estimates we have
(6.3)
|
∫
B
m
g
n
(
x
,
u
n
)
ϕ
n
d
x
-
∫
Γ
g
(
u
)
ϕ
d
x
′
|
≤
w
n
,
t
(
k
)
+
|
∫
B
m
g
n
(
x
,
T
k
(
u
n
)
)
ϕ
n
d
x
-
∫
Γ
g
(
T
k
(
u
)
)
ϕ
d
x
′
|
for some function wn,t(k) such that wn,t(k)→0 uniformly in t and n.
Fix now any ϵ>0 and g0∈C1(ℝ) such that sups∈[-k,k]|g0(s)-g(s)|≤ϵ. Then
|
∫
B
m
ϕ
n
ζ
n
(
x
N
)
g
(
T
k
(
u
n
)
)
d
x
-
∫
B
m
ϕ
n
ζ
n
(
x
N
)
g
0
(
T
k
(
u
n
)
)
d
x
|
≤
∥
ϕ
n
∥
∞
C
(
m
,
N
)
ϵ
and
|
∫
B
m
ϕ
ζ
n
(
x
N
)
g
(
T
k
(
u
)
)
d
x
-
∫
B
m
ϕ
ζ
n
(
x
N
)
g
0
(
T
k
(
u
)
)
d
x
|
≤
∥
ϕ
∥
∞
C
(
m
,
N
)
ϵ
.
Since g0∈C1(ℝ) has bounded derivative in [-k,k], we see that both g(Tk(u)) and g(Tk(un)) belong to W01,p(Bm)∩L∞(Bm). We note that we can integrate by parts for any pair of functions Ψ1∈W01,p(Bm)∩L∞(Bm) and Ψ2∈W1,p(Bm)∩L∞(Bm). We let
τ
n
(
x
)
:=
1
π
arctan
(
n
x
N
)
.
Then, using that ζn is smooth, we can write
∫
B
m
ζ
n
(
x
N
)
(
ϕ
n
g
0
(
T
k
(
u
n
)
)
d
x
-
ϕ
g
0
(
T
k
(
u
)
)
)
𝑑
x
=
-
(
A
)
-
(
B
)
,
where
(
A
)
=
∫
B
m
τ
n
[
(
∂
N
ϕ
n
)
g
0
(
T
k
(
u
n
)
)
-
(
∂
N
ϕ
)
g
0
(
T
k
(
u
)
)
]
𝑑
x
,
(
B
)
=
∫
B
m
τ
n
[
ϕ
n
g
0
′
(
T
k
(
u
n
)
)
∂
N
T
k
(
u
n
)
-
ϕ
g
0
′
(
T
k
(
u
)
)
∂
N
T
k
(
u
)
]
𝑑
x
.
Note that τn(t)→12(t|t|)=:τ(t) and g0(Tk(un))→g0(Tk(u)) strongly in Lr(Bm) for any 1≤r<∞ since τn, g0(Tk(un)), τ, and g0(Tk(u)) are uniformly bounded in L∞(Bm). Similarly τng0(Tk(un))→τg0(Tk(u)) strongly in Lp′(Bm). Thus, since (∂Nϕ)g0(Tk(u))∈Lp(Bm) and ∂Nϕn→∂Nϕ weakly in Lp(Bm), we conclude that (A) vanishes as n→∞ for any fixed k>0. We know that ϕn, ϕ, g0′(Tk(un)), and g0′(Tk(u)) are uniformly bounded in L∞(Bm). Hence, τnϕng0′(Tk(un))→τϕg0′(Tk(u)) strongly in Lp′(Bm). Given that ∂NTk(un)→∂NTk(u) weakly in Lp(Bm), we conclude that (B) vanishes as n→∞ for any fixed k>0. Collecting the above, we can rewrite (6.3) as
(6.4)
|
∫
B
m
g
n
(
x
,
u
n
)
ϕ
n
d
x
-
∫
Γ
g
(
u
)
ϕ
d
x
′
|
≤
w
n
,
t
(
k
)
+
w
k
,
ϵ
(
n
)
+
ϵ
C
(
m
,
N
)
(
∥
ϕ
n
∥
∞
+
∥
ϕ
∥
∞
)
+
|
∫
B
m
g
n
(
x
,
T
k
(
u
)
)
ϕ
d
x
-
∫
Γ
g
(
T
k
(
u
)
)
ϕ
d
x
′
|
for any ϵ>0, where wk,ϵ(n) is a function such that wk,ϵ(n)→0 as n→∞ for any k>0 and ϵ>0 fixed.
Given ϵ>0, we can find a closed set Ω0 such that u,ϕ∈C(Ω0) and cap1,p,N(Ω0c)<ϵ. Then, g(Tk(u)) and ϕ are uniformly continuous and bounded in Ω=Ω0∩Bm¯, and we can find t0 small enough so that
|
ϕ
(
x
′
,
x
N
)
g
(
T
k
(
u
)
)
(
x
′
,
x
N
)
-
ϕ
(
x
′
,
0
)
g
(
T
k
(
u
)
)
(
x
′
,
0
)
|
<
ϵ
for any (x′,xN)∈Ω∩{|xN|≤t0}. We can also assume t0 is such that
|
(
(
Γ
×
ℝ
)
∖
B
m
¯
)
∩
{
x
N
=
t
}
|
<
ϵ
for all |t|≤t0. Note that ∥χ{|t|>t0}ζn∥L1→0 as n→∞ for any t0>0. Then we write
|
∫
B
m
ϕ
g
n
(
x
,
T
k
(
u
)
)
d
x
-
∫
Γ
ϕ
g
(
T
k
(
u
)
)
d
x
′
|
=
|
∫
B
m
ϕ
g
n
(
x
,
T
k
(
u
)
)
d
x
-
∫
Γ
×
ℝ
ζ
n
(
x
N
)
(
ϕ
g
(
T
k
(
u
)
)
)
(
x
′
,
0
)
d
x
|
≤
∫
Γ
×
ℝ
ζ
n
(
x
N
)
|
ϕ
(
x
′
,
x
N
)
g
(
T
k
(
u
)
)
(
x
′
,
x
N
)
-
ϕ
(
x
′
,
0
)
g
(
T
k
(
u
)
)
(
x
′
,
0
)
|
𝑑
x
.
The above integrand can be estimated by ϵC(m,N) in Γ×ℝ∩(Ω∩{|xN|≤t0}), by ∥ϕ∥∞g~(k)∥χ{|xN|>t0}ζn∥L1 in Γ×ℝ∩({|xN|>t0}) and, in view of Lemma 6.3, by
∥
ϕ
∥
∞
g
~
(
k
)
∫
-
t
0
t
0
ζ
(
t
)
[
|
Ω
0
c
∩
{
x
N
=
t
}
|
+
|
(
(
Γ
×
ℝ
)
∖
B
m
¯
)
∩
{
x
N
=
t
}
|
]
d
t
≤
(
ϵ
1
p
C
(
m
,
N
,
p
)
+
ϵ
)
∥
ϕ
∥
∞
g
~
(
k
)
in Γ×ℝ∩(Ωc∩{|xN|≤t0}). Considering the above estimates, we obtain from (6.4) that
|
∫
B
m
g
n
(
x
,
u
n
)
ϕ
n
d
x
-
∫
Γ
g
(
u
)
ϕ
d
x
′
|
≤
w
n
,
t
(
k
)
+
w
~
k
,
ϵ
(
n
)
+
ϵ
1
p
C
(
m
,
N
,
p
,
g
~
(
k
)
,
∥
ϕ
∥
∞
,
∥
ϕ
n
∥
∞
)
for any 0<ϵ<1, where w~k,ϵ(n) is a function such that w~k,ϵ(n)→0 as n→∞ for any fixed k>0 and ϵ>0. Hence, the result follows. ∎
Lemma 6.7.
Fix m>0. Let un be renormalized solutions of
{
-
Δ
p
u
n
+
g
n
(
x
,
u
n
)
=
μ
¯
in
B
m
,
u
n
=
0
on
∂
B
m
,
where μ¯∈Mb(Bm), gn(x,s) is defined by (6.1), and the function g satisfies Assumption 1.3. Suppose un→u, ∇Tk(un)→∇Tk(u), and ∇un→∇u a.e. in Bm, where u satisfies condition (1) and (2) of Definition 3.1 and is cap1,p,N-q.e. finite. Assume also that |∇un|p-2∇un→|∇u|p-2∇u strongly in (Lq(Bm))N for any 1≤q<NN-1, Tk(un)→Tk(u) weakly in W01,p(Bm), and (6.2) holds. If g(u)∈L1(Bm∩∂R+N), then u is a renormalized solution of
{
-
Δ
p
u
+
g
(
u
)
ℋ
=
μ
¯
in
B
m
,
u
=
0
on
∂
B
m
.
Moreover, Tk(un)→Tk(u) strongly in W01,p(Bm) for any k>0.
Proof.
Repeat the argument in the proof of Lemma 4.2, replacing (4.1) by (6.2). Since the proof is almost the same, we omit it (see also [15, proof of Theorem 4.1]). ∎
The following lemma can be proven using the ideas in the proof of Lemma 6.6.
Lemma 6.8.
Let um and u be cap1,p,N-quasi-continuous functions such that um→u a.e. in RN and that Tk(um)→Tk(u) weakly in W1,p(BM) for any fixed k>0 and M∈N. Suppose g satisfies part (1) of Assumption 1.3 and let g~ be defined as in part (2) of Assumption 1.3. For any fixed M∈N we define
ρ
m
(
h
)
=
∫
B
M
∩
{
|
u
m
|
≥
h
}
∩
∂
ℝ
+
N
g
~
(
|
u
m
|
)
(
x
′
)
𝑑
x
′
,
ρ
(
h
)
=
∫
B
M
∩
{
|
u
|
≥
h
}
∩
∂
ℝ
+
N
g
~
(
|
u
|
)
(
x
′
)
𝑑
x
′
.
If ρm(h)→0 and ρ(h)→0 as h→∞, uniformly in m, then (4.1) holds in BM with gm=g(um) and g=g(u).
We now prove Theorem 1.4 .
Proof of Theorem 1.4.
We consider two cases.
Case 𝟏<p<N. We divide the proof into four steps.
Step 1: Definition of umn and um. Let gn(x,s) be defined by (6.1) and fix m∈ℕ. By [23, Theorem 5.1.2] for any n∈ℕ there exists a renormalized solution of
(6.5)
{
-
Δ
p
u
m
n
+
2
g
n
(
x
,
u
m
n
)
=
2
μ
m
in
B
m
,
u
m
n
=
0
on
∂
B
m
,
where μm is the restriction of μ to Bm. By writing μ~mn=μm-gn(x,umn), we see that umn is a renormalized solution to -Δpumn=2μ~mn in Bm. By Proposition 6.2 we have |μ~mn|(Bm)≤2|μ|(ℝN)<∞ and so we can apply Theorem 3.3 to obtain that, passing to subsequences, umn→um a.e. in Bm as n→∞ for suitable behaved functions um. Since each umn satisfies the estimate ∥∇Tk(umn)∥pp≤2k|μ~mn|(Bm)≤4k|μ|(ℝN), so does the functions um.
Step 2: Convergence of gn(x,umn) for fixed m∈N. Fix |t|<m and for any n and h write Enh={|umn|<h} and Eh={|um|<h}. Define σ(s)=|{x∈Bm∩{xN=t}:|umn|>s}|. Proceeding as in Remark 3.5, we see that
∫
B
m
∩
(
E
n
h
)
c
∩
{
x
N
=
t
}
|
g
(
u
m
n
)
|
(
x
′
,
t
)
𝑑
x
′
≤
σ
(
h
)
g
~
(
h
)
+
∫
h
∞
σ
(
s
)
𝑑
g
~
(
s
)
.
Combining Lemma 6.4, integration by parts, and Assumption 1.3, we obtain that this last quantity is bounded by
C
(
N
,
p
,
m
)
∥
μ
m
∥
𝔐
b
N
-
1
N
-
p
∫
h
∞
g
~
(
s
)
s
-
p
(
N
-
2
)
+
1
N
-
p
𝑑
s
which vanishes as h→∞, independently of t and n. Applying the same argument to the functions um, we conclude
∫
B
m
∩
(
E
n
h
)
c
∩
{
x
N
=
t
}
|
g
(
u
m
n
)
|
(
x
′
,
t
)
𝑑
x
′
+
∫
∂
ℝ
+
N
∩
B
m
∩
(
E
h
)
c
|
g
(
u
m
)
|
𝑑
x
′
→
0
as h→∞, uniformly in t and n. Note also that the argument implies g(um)∈L1(Bm∩∂ℝ+N). Hence, we can combine Lemma 6.6 and Lemma 6.7 to obtain that um is a renormalized solution of
{
-
Δ
p
u
m
=
2
μ
m
-
2
g
(
u
m
)
ℋ
in
B
m
,
u
m
=
0
on
∂
B
m
.
Moreover, by Proposition 6.2 and (6.2) we get that g(um) is uniformly bounded in L1(∂ℝ+N∩Bm). Thus, we can apply Lemma 4.1 with data 2μm-2g(um)ℋ to obtain a suitable limit function u such that um→u a.e. in ℝN.
Step 3: Estimates on the level sets of um(x′,0) and u(x′,0). Fix any M>0. Since um satisfies estimate (3.3) and ∥μm-g(um)ℋ∥𝔐b is uniformly bounded, we can find k0(M,N,p,μ) independent of m such that |{|um|>k2}|≤|BM|4 for all k≥k0. Let ck,m,M=(Tk(um))M be the average of Tk(um) in BM. Then |ck,m,M|≤3k4 for all k≥k0. As in the proof of Lemma 6.4, replacing Poincaré inequality with Poincaré–Wirtinger inequality, we deduce
|
{
|
u
m
|
≥
k
}
∩
B
M
∩
∂
ℝ
+
N
|
≤
C
(
N
,
p
,
B
M
,
∥
μ
∥
𝔐
b
)
k
-
(
p
-
1
)
(
N
-
1
)
N
-
p
.
Similarly, by Fatou’s Lemma, u satisfies estimate (3.2) in BM, while if ck,M=(Tk(u))M is the average of Tk(u) in BM, then |ck,M|≤3k4 for k≥k0 (see also the proof of Lemma 4.1). Thus, we have the same estimate for |{|u|≥k}∩BM∩∂ℝ+N|.
Step 4: Passing to the limit in m. As above, by the assumptions on g~ and the decay estimates on um and u, we can apply Lemma 6.8 to obtain condition (4.1) (with gm=g(um) and g=g(u)). Note that from (4.1) we conclude that ∥g(u)∥L1(∂ℝ+N)≤∥μ∥𝔐b. Then Lemma 4.2 implies that u is a local renormalized solution of -Δpu=2μ-2g(u)ℋ in ℝN. Since the measures 2μm-2g(um)ℋ are supported in ∂ℝ+N, we apply Theorem 5.1 to obtain that um, and thus u, are symmetric with respect to ∂ℝ+N. Hence, by Theorem 5.2 the restriction of u to ℝ+N is a solution to the problem.
Case p=N. We repeat the ideas used in the case 1<p<N, so we only point out the main differences. To repeat Step 1, we need to obtain solutions umn to (6.5). Theorem 5.1.2 of [23] guarantees the existence of such solutions provided ∥μ∥𝔐b is bounded by (C0Nγ)N-1, where C0=C0(Bm) is a constant that may depend on the domain Bm. However, [23, proof of Theorem 5.1.2] shows that the constant C0 is exactly the constant in the estimate
|
{
x
∈
B
m
:
|
u
ϵ
|
>
k
}
|
≤
C
(
N
,
B
m
)
e
-
C
0
k
∥
μ
∥
𝔐
b
1
1
-
N
,
which holds for solutions uϵ to problem (1.2) with μ replaced by a regularized μϵ. Since any such solution satisfies Tk(uϵ)∈W01,N(Bm) and ∥∇Tk(uϵ)∥N≤(Ck∥μ∥𝔐b)1N, we can use the results of [6] to replace the constant C0 in above estimate with some constant c1(N) independent of Bm (see Remark 6.5). Hence, using the above estimate, we see that in fact a solution umn exists provided ∥2μm∥𝔐b≤(c1Nγ)N-1. We also note that Step 2 can be repeated without any major modification under the same boundedness condition on the measure.
Step 3 can be obtained as follows. By the results of [10] the solutions um belong the Lorentz–Sobolev space W1LN,∞(Bm) and satisfy
∥
∇
u
m
∥
L
N
,
∞
(
B
m
)
≤
C
(
N
)
∥
2
μ
∥
𝔐
b
1
N
-
1
.
In particular,
∥
u
m
∥
BMO
(
B
M
)
≤
C
(
N
,
B
M
)
∥
2
μ
∥
𝔐
b
1
N
-
1
for any m≥M (see also [7]). Here BMO(BM) is the space of LN(BM) functions of bounded mean oscillation (see [10] for a definition of BMO). By [6, Theorem 2.5], we can assert
∫
B
M
∩
∂
ℝ
+
N
e
C
1
|
u
m
-
(
u
m
)
M
|
∥
∇
u
m
∥
L
N
,
∞
(
B
m
)
𝑑
x
′
≤
C
(
B
M
,
N
)
for some constant C1(N) and where (um)M is the average of um in BM. Hence, as in the case 1<p<N, we follow the proof of Lemma 6.4 together with
{
|
u
m
|
≥
k
}
⊂
{
|
u
m
-
(
u
m
)
M
|
≥
k
-
|
(
u
m
)
M
|
}
and
|
(
u
m
)
M
|
≤
C
(
N
)
∥
u
m
∥
BMO
(
B
M
)
to get
|
{
x
∈
B
M
∩
∂
ℝ
+
N
:
|
u
m
|
>
k
}
|
≤
C
(
N
,
B
M
,
μ
)
e
-
c
2
k
∥
2
μ
∥
𝔐
b
1
1
-
N
with c2=c2(N). Thus
∫
B
M
∩
{
|
u
m
|
≥
k
}
∩
∂
ℝ
+
N
g
~
(
|
u
m
|
)
𝑑
x
′
≤
C
(
N
,
B
M
,
μ
)
∫
k
∞
g
~
(
s
)
e
-
c
2
s
∥
2
μ
∥
𝔐
b
1
1
-
N
𝑑
s
,
which vanishes as k→∞ provided ∥2μ∥𝔐b≤(c2Nγ)N-1. Combining the definition of the LN,∞(BM) norm (see [17]), Fatou’s Lemma, and density, one can show that ∥∇u∥LN,∞(BM) satisfies the same estimate. Hence, Step 4 of the first case can be applied and we obtain a solution to the problem provided 2∥μ∥𝔐b≤(cNγ)N-1, where c=c(N)=min(c1(N),c2(N)). ∎
Note that we have proven stability of solutions without using any type of convergence of umn to um (or of um to u) in ∂ℝ+N. Combining Lemma 6.7 and [9, Proposition 2.3.8], we may pass to a diagonal subsequence, which we relabel as {umn}n, such that Tk(umn)→Tk(um)cap1,p,N-q.e. in Bm as n→∞ for any k∈ℕ. Hence umn→um a.e. in Bm∩∂ℝ+N. By Vitali’s Theorem it follows that g(umn)→g(um) strongly in L1(Bm∩∂ℝ+N). Similarly, Lemma 4.2 shows that, up to a subsequence, um→u a.e. in ∂ℝ+N and g(um)→g(u) strongly in L1(∂ℝ+N∩BM) for any M∈ℕ.
7 Nonlinear Problems with Absorption. The Supercritical Case
Throughout this subsection we will assume that
g
:
ℝ
→
ℝ
is a continuous nondecreasing odd function and
1
<
p
<
N
.
We will use the existence results for problem (1.2) obtained in [4] (see also [23]). The main tool we use is the R-truncated Wolff potential of a nonnegative measure μ∈𝔐b(ℝN), which is defined by
W
α
,
s
,
N
R
[
μ
]
(
x
)
=
∫
0
R
(
μ
(
B
t
(
x
)
)
t
N
-
α
s
)
1
s
-
1
d
t
t
,
where α>0, 1<s<α-1N, 0<R≤∞, and Bt(x) is the N-dimensional ball of radius t centered at x∈ℝN. If R=∞, we drop it from the notation. Note that
(7.1)
W
1
,
p
,
N
R
[
μ
]
(
x
′
,
x
N
)
≤
W
1
-
1
/
p
,
p
,
N
-
1
R
[
μ
]
(
x
′
)
for any (x′,xN)∈ℝN, and if μ is supported in ∂ℝ+N, then
(7.2)
W
1
-
1
/
p
,
p
,
N
-
1
R
[
μ
]
(
x
′
)
=
W
1
,
p
,
N
R
[
μ
]
(
x
′
,
0
)
.
Remark 7.1.
Let us record the following important relationship between Wolff potential and p-superharmonic functions: if u is a nonnegative p-superharmonic function in Ω, 1<p≤N, and -Δpu=μ in Ω, then there exist positive constants c1, c2, c3, depending only on p and N, such that for any x∈Ω and B3r(x)⊂⊂Ω there holds
c
1
W
1
,
p
,
N
r
[
μ
]
(
x
)
≤
u
(
x
)
≤
c
2
inf
B
r
(
x
)
u
+
c
3
W
1
,
p
,
N
r
[
μ
]
(
x
)
(see [12, 16, 23]).
Our first result is an improvement on estimates obtained in [4] for problem (1.2).
Lemma 7.2.
Let Ω be a bounded domain and let g:Ω×R→R be a Carathéodory function such that s↦g(x,s) is nondecreasing and odd for a.e. x∈Ω. Let μ∈Mb(Ω) and let u be a renormalized solution to (1.2). Then
-
c
W
1
,
p
,
N
2
diam
(
Ω
)
[
μ
-
]
(
x
)
≤
u
(
x
)
≤
c
W
1
,
p
,
N
2
diam
(
Ω
)
[
μ
+
]
(
x
)
cap
1
,
p
,
N
-
q.e. in Ω with c=c(N,p) the same constant as in [4, Theorem 4.1].
Proof.
We know that for every k≥0 the functions Tk(u)=uk are renormalized solutions to
{
-
Δ
p
u
k
+
g
(
x
,
u
k
)
χ
{
|
u
|
<
k
}
=
μ
0
χ
{
|
u
|
<
k
}
+
λ
k
+
-
λ
k
-
in
Ω
,
u
=
0
on
∂
Ω
,
for some nonnegative measures λk±∈𝔐b(Ω) that converge to μs± in the narrow topology of measures (see [8]). Let vk be a renormalized solution to
{
-
Δ
p
v
k
=
μ
0
+
χ
{
|
u
|
<
k
}
+
λ
k
+
in
Ω
,
v
k
=
0
on
∂
Ω
.
Since μ0+χ{|u|<k}+λk+ is nonnegative, we have vk≥0 (see [16, Remark 6.5]), and so g(x,vk)≥0 a.e. in Ω. Since uk is bounded, we have g(x,vk)χ{uk>vk}∈L1(Ω) and so we can use that g(x,s) is a.e. nondecreasing on s and that all the measures involved are in 𝔐0(Ω) to obtain, by an easy adaptation of the proof of Lemma 6.8 of [16], that uk≤vk a.e. in Ω. By [8, Theorem 3.4], passing to a subsequence we have vk→v a.e. in Ω, where v is a renormalized solution to
{
-
Δ
p
v
=
μ
0
+
+
μ
s
+
=
μ
+
in
Ω
,
v
=
0
on
∂
Ω
.
In particular, since u is a.e. finite, u≤v a.e. in Ω.
Since μ+ is nonnegative, by [16, Theorem 2.1], v coincides a.e. in Ω with a p-superharmonic function v~ satisfying
v
~
(
x
)
≤
c
W
1
,
p
,
N
2
diam
(
Ω
)
[
μ
+
]
(
x
)
in Ω, where c=c(N,p) (see [4, proof of Theorem 3.8]). Moreover, by [12, Theorem 10.9], v~ is cap1,p,N-quasi-continuous in Ω. Considering the cap1,p,N-quasi-continuous representative of u, and since u≤v~ a.e. in Ω, we can conclude u≤v~cap1,p,N-q.e. in Ω. Hence, the above estimate holds with u in place of v~, cap1,p,N-q.e. in Ω. The lower estimate can be obtained similarly. ∎
Remark 7.3.
We note that in the second part of the above proof we have used that the p-superharmonic representative of a nonnegative renormalized solutions u is a cap1,p,N-quasi-continuous representative of u. Let us also mention the following: if u≤v a.e. in Ω, where u and v are p-superharmonic, then u≤v everywhere in Ω (see [12, Corollary 7.23]).
The following lemma asserts that for a nonnegative measure it is possible to obtain nondecreasing solutions defined on nondecreasing domains. It can be proven following the ideas in [4, Lemmas 4.2 and 4.3].
Lemma 7.4.
Let Ω and Ω′ be bounded domains such that Ω⊂Ω′. Let μ∈Mb(Ω) be nonnegative, compactly supported in Ω, and let R=2diam(Ω′). Assume g(x,cW1,p,NR[μ](x)) is in L1(Ω′), where g(x,s) is a Carathéodory function such that s↦g(x,s) is nondecreasing and odd for a.e. x∈Ω, and c=c(N,p) is as in Lemma 7.2. Then there exist renormalized solutions u and v to (1.2), in Ω and Ω′, respectively, such that u≤v a.e. in Ω.
Next we use Lemma 7.2 to show that we can obtain solutions to (1.4) from solutions to problem (1.2).
Lemma 7.5.
Let g be a continuous nondecreasing odd function and let c=c(N,p) be the constant in Lemma 7.2. Let μ∈Mb(∂R+N) be such that g∘cW1-1/p,p,N-14m[μ±] is in L1(∂R+N∩Bm). Let gn(x,s) be defined as in (6.1) and let un be renormalized solutions to
{
-
Δ
p
u
n
+
g
n
(
x
,
u
n
)
=
μ
in
B
m
,
u
n
=
0
on
∂
B
m
.
Then there exists a function u and a subsequence of {un}n, which we relabel as {un}n, such that un→u a.e. in Bm and u is a renormalized solution to (1.4) in Ω=Bm that satisfies
(7.3)
-
c
W
1
-
1
/
p
,
p
,
N
-
1
4
m
[
μ
-
]
(
x
′
)
≤
u
(
x
′
,
x
N
)
≤
c
W
1
-
1
/
p
,
p
,
N
-
1
4
m
[
μ
+
]
(
x
′
)
cap
1
,
p
,
N
-
q.e in Bm.
Proof.
By Lemma 7.2 and by (7.1), the functions un satisfy (7.3). Note that by Proposition 6.2 the measures μ~n=μ-gn(x,un) are bounded independent of n, and so we can apply Theorem 3.3 to obtain that, up to a subsequence, there exists a suitable behaved function u defined in Bm such that un→u a.e. in Bm as n→∞.
Now, to pass to the limit in the equation solved by un, we repeat Step 2 in the proof of Theorem 1.4. Using than un satisfies (7.3), which holds a.e. in the intersection of Bm with any hyperplane, that g is nondecreasing, and that g(cW1-1/p,p,N-14m[μ±])∈L1(Bm∩∂ℝ+N), we can obtain the necessary decay estimates for the functions un. To extend the estimates to the limit function u, we need to show that u also satisfies (7.3). Looking at the proof of Lemma 7.2, we see that we obtain the right-hand side of estimate (7.3) for un from the inequality un≤v for some particular renormalized solution v. Using that u is the a.e. limit of the un, we get u≤v a.e. in Bm. Then, considering cap1,p,N-quasi-continuous representatives, we conclude u≤vcap1,p,N-q.e. in Bm and so, proceeding as in the proof of the lemma, we obtain that the right-hand side of estimate (7.3) also holds for u. The left-hand side estimate follows in the same way, and so (7.3) holds for u. ∎
We are now ready to show the following trace version of [4, Theorem 4.1].
Theorem 7.6.
Let g be a continuous nondecreasing odd function, and let c1=21p-1c(N,p), where c(N,p) is the constant in Lemma 7.2. Assume μi∈Mb(∂R+N), i=1,2, are nonnegative and for every m∈N there exist nondecreasing sequences {μi,km}k of nonnegative measures in Mb(∂R+N) with compact support in Bm∩∂R+N converging to μim=.μi|Bm∩∂R+N weakly-∗ in Mb(Bm∩∂R+N) such that g∘c1W1-1/p,p,N-14(m+1)[μi,km] is in L1(∂R+N∩Bm) and μi,km≤μi,km+1 for each k∈N. Then there exists a renormalized solution of
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
+
g
(
u
)
=
μ
1
-
μ
2
on
∂
ℝ
+
N
.
Moreover,
(7.4)
-
c
1
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
2
]
(
x
′
)
≤
u
(
x
′
,
x
N
)
≤
c
1
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
1
]
(
x
′
)
cap
1
,
p
,
N
-
q.e. in R+N¯.
Proof.
Let gn(x,s) be defined as in (6.1). Since gn(x,s)=ζn(xN)g(s) satisfies the assumptions of [4, Theorem 4.1] we may apply [4, Lemma 4.3] to obtain renormalized solutions umn,k and um,in,k, with i=1,2, of
{
-
Δ
p
u
+
2
g
n
(
x
,
u
)
=
2
μ
in
B
m
,
u
=
0
on
∂
B
m
,
with data μ=μ1,km-μ2,km, and μ=μi,km, respectively, satisfying -um,2n,k≤umn,k≤um,1n,k a.e. in Bm. By the same lemma, we can assume um,in,k are nondecreasing in k, while by [16, Remark 6.5] we know they are nonnegative. By combining [4, Lemma 4.3] and Lemma 7.4 above we may further assume they are nondecreasing in m.
Step 1: Limit as n→∞. We apply Lemma 7.5 to take limit in n and obtain renormalized solutions umk, and um,ik of (1.4). By Lemma 6.7 we have Th(um,in,k)→Th(um,ik) and Th(umn,k)→Th(umk) strongly in W01,p(Bm) for any h>0. Hence, by [9, Proposition 2.3.8], we might pass to a diagonal subsequence so that um,in,k→um,ik and umn,k→umkcap1,p,N-q.e. in Bm (see the comments at the end of Section 6). Hence, we can assume that
(7.5)
-
c
W
1
-
1
/
p
,
p
,
N
-
1
4
m
[
2
μ
2
,
k
m
]
(
x
′
)
≤
-
u
m
,
2
k
(
x
′
,
t
)
≤
u
m
k
(
x
′
,
t
)
≤
u
m
,
1
k
(
x
′
,
t
)
≤
c
W
1
-
1
/
p
,
p
,
N
-
1
4
m
[
2
μ
1
,
k
m
]
(
x
′
)
cap
1
,
p
,
N
-q.e. in ℝN. Note also that the limit functions um,ik are nonnegative and nondecreasing in k and mcap1,p,N-q.e., and so in particular a.e. in Bm∩∂ℝ+N.
Step 2: Limit as k→∞. Since the measures μi,km are uniformly bounded in norm by μi, by Proposition 6.2, (6.2), and (7.5), we get
∥
g
(
u
m
,
i
k
)
∥
L
1
(
B
m
∩
∂
ℝ
+
N
)
+
∥
g
(
u
m
k
)
∥
L
1
(
B
m
∩
∂
ℝ
+
N
)
≤
2
∥
μ
1
∥
𝔐
b
+
2
∥
μ
2
∥
𝔐
b
.
Then we apply Theorem 3.3 to, up to subsequences, pass to the limit in k and obtain limit functions um and um,i. There is no loss of generality in assuming um,ik coincides with its p-superharmonic representative mentioned in Remark 7.3, so that in particular we can assume um,ik are nondecreasing in k everywhere in Bm. Then [12, Lemma 7.3] shows that supkum,ik is a p-superharmonic function in Bm and so, by [12, Theorem 10.9], also cap1,p,N-quasi-continuous in Bm. Hence, supkum,ik is a cap1,p,N-quasi-continuous representative of um,i, and we can assume um,ik→um,icap1,p,N-q.e. in Bm. Thus, we can extend (7.5) to the limit functions.
Since we have the L1(Bm∩∂ℝ+N) estimate on g(um,ik), and um,ik are nondecreasing in k and nonnegative, we obtain g(um,ik)→g(um,i)∈L1(Bm∩∂ℝ+N) and extend the estimates to ∥g(um,i)∥L1(Bm∩∂ℝ+N). Then, by slightly modifying the arguments leading to [4, Corollary 3.5] we obtain that um,i are renormalized solutions to (1.4) with data μ=μim. Indeed, to obtain the same stability result we only need to consider the terms g(um,ik)ℋ and g(um,i)ℋ, since the focus of the corollary is handling the measures μm,ik in order to apply the stability result of [8]. But, replacing this stability result by the one in [15], we see by the proof of Lemma 6.7 (or Lemma 4.2) that we can prove stability provided (4.1) holds in Bm. To this end, Lemma 6.8 can be applied using that um,ik are nonnegative and g(um,ik)↑g(um,i)∈L1(Bm∩∂ℝ+N). A similar argument shows that um is a renormalized solution to (1.4) with data μ=μ1m-μ2m. This follows from the hypothesis on g and (7.5).
Step 3: Limit as n→∞. We use the uniform bounds on the L1(Bm∩∂ℝ+N) norm of g(um) to apply Lemma 4.1 and obtain a function u as the limit of the um. Note that we can also take the limit of the um,i to obtain functions ui. As we argued in the previous step, using that um,i are nondecreasing in m and passing to cap1,p,N-quasi-continuous representatives, we extend (7.5) to the limit functions. Using this estimates and the hypothesis on g, we obtain uniform bounds on the L1(Bm∩∂ℝ+N) norms of g(ui). Then Lemma 6.8 applies and g(u)∈L1(∂ℝ+N). Thus, by Lemma 4.2 we obtain -Δpu+2g(u)ℋ=2μ in ℝN. Applying Theorem 5.1 and Theorem 5.2, we obtain that the restriction of u to ℝ+N is a solution of the desired problem satisfying (7.4). ∎
As in Section 6, we observe that it can be shown that, passing to subsequences, g(umn,k)→g(umk) and g(umk)→g(um) strongly in L1(Bm∩∂ℝ+N), and g(um)→g(u) strongly in L1(∂ℝ+N∩BM) for any M∈ℕ.
Proof of Theorem 1.5.
Note that μ1=μ+, μ2=μ-, and μim=.μi|Bm, i=1,2, are absolutely continuous with respect to capp-1,q/(q-p+1),N-1 . Then for every m we can apply [4, Theorem 2.6] in dimension N-1 with s1=s2=q, α=1-1p, and R=4(m+1) to obtain nondecreasing sequences {μi,km}k of nonnegative measures in 𝔐b(∂ℝ+N) satisfying the necessary conditions to apply Theorem 7.6. Looking at the [4, proof of Theorem 2.6] we see it can be assumed that μi,km≤μi,km+1 for each k∈ℕ. ∎
For 1≤s1<∞ and 1<s2≤∞ we denote by Ls1,s2(ℝN) the standard Lorentz space (see for example [17]). For α>0 one can define the Lorentz–Bessel capacities
cap
α
,
s
1
,
s
2
,
N
(
E
)
=
inf
{
∥
f
∥
L
s
1
,
s
2
(
ℝ
N
)
:
f
≥
0
,
𝒢
α
∗
f
≥
1
on
E
}
(see [9] or [4]). Then, repeating the ideas in the previous proof, it is possible to show the following result.
Theorem 7.7.
Let 1<p<N and let g be a continuous nondecreasing odd function such that
∫
1
∞
g
(
s
)
s
-
(
q
+
1
)
𝑑
s
<
∞
for some q>p-1. If μ∈Mb(∂R+N) is absolutely continuous with respect to capp-1,q/(q-p+1),1,N-1, then there exists a renormalized solution to (1.1) with datum μ.
Let us also mention that it is possible to prove a trace version of [4, Theorem 1.2].
8 Nonlinear Problems with Source
We recall the Riesz potentialIα,N of order α, 0<α<N, on ℝN, of a nonnegative Radon measure μ is
I
α
,
N
[
μ
]
(
x
)
=
c
(
N
,
α
)
∫
ℝ
N
|
x
-
y
|
α
-
N
𝑑
μ
(
y
)
,
where c(N,α) is a normalized constant.
Theorem 8.1.
Let 1<p<N and p-1<q. Let μ in Mb(∂R+N) be nonnegative and suppose there exists a nonnegative renormalized solution to (1.5). Then
(8.1)
∫
B
(
I
p
-
1
,
N
-
1
[
μ
B
]
)
q
p
-
1
𝑑
x
′
≤
C
(
N
,
p
,
q
)
μ
(
B
)
holds for all balls B⊂∂R+N≃RN-1 (where μB is the restriction of μ to B). Moreover, the above condition implies W1-1/p,p,N-1[μ]∈Lq(∂R+N) and
(8.2)
W
1
-
1
/
p
,
p
,
N
-
1
[
(
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
)
q
]
≤
C
1
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
a.e. in
∂
ℝ
+
N
for some nonnegative constant C1 depending on p, q, and N.
Proof.
If u solves (1.5), then u¯, the extension of u to ℝN by even reflection across ∂ℝ+N, is a local renormalized solution to -Δpu¯=2u¯qℋ+2μ in ℝN. Let ω=2u¯qℋ+2μ. Combining Theorems 4.3.2 and 4.2.5 of [23], we obtain that u¯ coincides a.e. with a p-superharmonic function u~ satisfying W1,p,N[ω]≤C(N,p)u~. By Remark 7.3 we can conclude that u¯ satisfies the same estimate cap1,p,N-q.e. in ℝN and so, by Proposition 2.2, ℋ-a.e. Thus, for any dyadic cube P⊂∂ℝ+N (i.e., P=2j(k+[0,1)N-1) for some j∈ℤ and k∈ℤN-1) we have
ω
(
P
)
≥
∫
P
2
u
¯
q
𝑑
x
′
≥
C
∫
P
W
1
,
p
,
N
[
ω
]
q
𝑑
x
′
=
C
∫
P
W
1
-
1
/
p
,
p
,
N
-
1
[
ω
]
q
𝑑
x
′
.
From this point forward we can proceed as in [16, proof of Theorem 2.3], using the dyadic representation of the Wolff and Riesz potentials, [16, Proposition 3.1], and [20, Theorem 3], to obtain (8.1).
To prove the second assertion, we can proceed as in [16], using the results in [22] and [16, Proposition 5.1], to show that (8.1) implies
(8.3)
∫
ℝ
N
-
1
(
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
B
]
)
q
𝑑
x
′
≤
C
(
N
,
p
,
q
)
μ
(
B
)
for all balls B⊂∂ℝ+N≃ℝN-1. Note that, by monotone convergence, the above condition now gives that if μ∈𝔐b(∂ℝ+N), then W1-1/p,p,N-1[μ]∈Lq(∂ℝ+N). Finally, we want to show that (8.3) implies (8.2). In view of (7.2), the proof of Theorem 2.3 in [16] can be repeated in our setting without any further modification. ∎
We now obtain solutions to problem (1.5). We define a recursive sequence of solutions to
(8.4)
{
-
Δ
p
u
m
=
2
u
m
-
1
q
ℋ
+
2
μ
m
in
B
m
,
u
m
=
0
on
∂
B
m
,
where u0=0 and μm(E)=μ(E∩Bm).
Theorem 8.2.
Let p and q with 1<p<N and p-1<q. Let μ be a nonnegative measure in Mb(∂R+N) satisfying W1-1/p,p,N-1[μ]∈Lq(∂R+N) and condition (8.2) with
C
1
≤
(
q
-
p
+
1
q
c
(
N
,
p
)
C
(
p
)
2
1
p
-
1
)
q
p
-
1
(
p
-
1
q
-
p
+
1
)
,
where C(p)=max{1,22-pp-1} and c(N,p) is the constant in Lemma 7.2. Then there exists a nonnegative renormalized solution to (1.5) satisfying
u
(
x
′
,
x
N
)
≤
(
q
c
(
N
,
p
)
C
(
p
)
2
1
p
-
1
q
-
p
+
1
)
W
1
-
1
/
p
,
p
,
N
-
1
[
μ
]
(
x
′
)
in
Ω
∩
ℝ
+
N
¯
,
where Ω is a set of the form Ω=Ω1∩(Ω2×R), Ω2⊂RN-1, with cap1,p,N(Ω1c)=0 and |Ω2c|=0. In particular, the above estimate holds a.e. in any hyperplane RN-1×{t}.
Proof.
Let u1 be a renormalized solution of (8.4). Since μ is nonnegative so is u1 (see [16, Remark 6.5]), and by Lemma 7.2, (7.2), and (7.1) we get u1∈Lq(B1∩∂ℝ+N) and
u
1
(
x
′
,
x
N
)
≤
c
(
N
,
p
)
W
1
-
1
/
p
,
p
,
N
-
1
[
2
μ
]
(
x
′
)
cap
1
,
p
,
N
-q.e. in B1, where c(N,p) is the constant in Lemma 7.2.
Suppose m∈ℕ, m>1, and um is a renormalized solution to problem (8.4), where μm(E)=μ(E∩Bm) and um-1∈Lq(∂ℝ+N) is nonnegative, supported in Bm-1, and satisfies
u
m
-
1
(
x
′
,
x
N
)
≤
α
m
-
1
W
1
-
1
/
p
,
p
,
N
-
1
[
2
μ
]
(
x
′
)
for every
(
x
′
,
x
N
)
∈
Ω
∩
B
m
-
1
for some constant αm-1, where Ω is a set of the form Ω=Ω1,m-1∩(Ω2×ℝ) with cap1,p,N(Ω1,m-1c)=0 and Ω2⊂ℝN-1 is the set where W1-1/p,p,N-1[μ](x′) is finite and condition (8.2) holds (note that |Ω2c|=0). Since the measure 2um-1qℋ+2μm is nonnegative, we have um≥0 and, again by Lemma 7.2,
u
m
≤
c
(
N
,
p
)
W
1
,
p
,
N
4
m
[
2
u
m
-
1
q
ℋ
+
2
μ
m
]
cap
1
,
p
,
N
-q.e. in
B
m
.
By the definition of the Wolff potential one can see that
W
1
,
p
,
N
4
m
[
2
u
m
-
1
q
ℋ
+
2
μ
m
]
≤
C
(
p
)
(
W
1
,
p
,
N
4
m
[
2
u
m
-
1
q
ℋ
]
+
W
1
,
p
,
N
4
m
[
2
μ
m
]
)
,
where C(p)=max{1,22-pp-1}. Therefore, we can use that um-1qℋ and μm are supported in ∂ℝ+N, together with (7.1), the monotonicity of the Wolff potential, assumption (8.2), and the induction hypothesis, to obtain
u
m
(
x
′
,
x
N
)
≤
C
(
p
)
c
(
N
,
p
)
(
(
2
1
p
-
1
α
m
-
1
)
q
p
-
1
C
1
+
1
)
W
1
-
1
/
p
,
p
,
N
-
1
[
2
μ
]
(
x
′
)
for every (x′,xN)∈Ω∩Bm, where Ω=Ω1,m∩(Ω2×ℝ), Ω2⊂ℝN-1 is as described above, and Ω1,m is the intersection of Ω1,m-1 with the set where the first inequality holds. Note that cap1,p,N(Ω1,mc)=0. Hence, by induction starting with α1=c(N,p), we obtain a sequence of nonnegative functions {um}m⊂Lq(∂ℝ+N) such that
u
m
(
x
′
,
x
N
)
≤
α
m
W
1
-
1
/
p
,
p
,
N
-
1
[
2
μ
]
(
x
′
)
in
Ω
∩
B
m
,
with Ω as described above, and where
α
m
=
C
(
p
)
c
(
N
,
p
)
(
(
2
1
p
-
1
α
m
-
1
)
q
p
-
1
C
1
+
1
)
.
Since C(p)≥1, it is easy to show by induction that the assumption on C1 implies that the sequence {αm}m satisfies
α
m
≤
M
:=
q
c
(
N
,
p
)
C
(
p
)
q
-
p
+
1
for all
m
∈
ℕ
.
Note that we may assume um-1≤umcap1,p,N-q.e. in ℝN. Indeed, let um-1 be a solution of (8.4) such that um-2≤um-1cap1,p,N-q.e. in ℝN. Set νm=2um-1ℋ+μm. Then, since νm-1≤νm, an easy adaptation of [16, Lemma 6.9] shows that we can obtain a renormalized solution um of (8.4) such that um-1≤um a.e. in Bm-1. Extending by zero and using cap1,p,N-quasi-continuous representatives, we conclude um-1≤umcap1,p,N-q.e. in ℝN.
Now, since these solutions are nonnegative, we may identify them with their p-superharmonic representatives and conclude um-1≤um everywhere in ℝN (see Remark 7.3). Then, by [12, Lemma 7.3], u=supmum defines a p-superharmonic function which, by [12, Theorem 10.9], is cap1,p,N-quasi-continuous in ℝN (note that u is finite in Ω and |Ωc∩Bm|=0 for every m∈ℕ). Moreover, it follows that u∈Lq(∂ℝ+N) and umq→uq in L1(∂ℝ+N). Notice that {um}m is uniformly bounded in Lq(∂ℝ+N). Hence, by Lemma 4.1u satisfies properties (1), (2), and (3) in the statement of the said lemma. To show that (4.1) holds, we use that um↑u a.e. in BM∩∂ℝ+N and uq∈L1(∂ℝ+N) and apply Lemma 6.8. We finish by combining Lemma 4.2, Theorem 5.1, and Theorem 5.2. ∎
Proof of Theorem 1.6.
We know (1) implies (2) from Theorem 8.1. If (2) holds for some constant C, then for any ϵ>0,
W
1
-
1
/
p
,
p
,
N
-
1
[
(
W
1
-
1
/
p
,
p
,
N
-
1
[
ϵ
μ
]
)
q
]
≤
C
ϵ
q
-
(
p
-
1
)
(
p
-
1
)
2
W
1
-
1
/
p
,
p
,
N
-
1
[
ϵ
μ
]
a.e. in ∂ℝ+N, and so (1) follows from Theorem 8.2 provided ϵ>0 is chosen small enough. ∎
We now turn to the problem of nonexistence. Following [16, proof of Theorem 2.3], one can actually show
∫
B
(
I
p
-
1
,
N
-
1
[
ω
B
]
)
q
p
-
1
𝑑
x
′
≤
C
(
N
,
p
,
q
)
ω
(
B
)
,
where ω=2u¯qℋ+2μ. Note also that the argument can be applied directly to a p-superharmonic function v solving -Δpv=2vqℋ+2μ in ℝN. It also follows from [16, proof of Theorem 2.3] that
∑
Q
⊂
P
(
ω
(
Q
)
|
Q
|
1
-
p
-
1
N
-
1
)
q
p
-
1
|
Q
|
≤
C
(
N
,
p
)
ω
(
P
)
for all dyadic cubes Q,P⊂∂ℝ+N, which in the case p=N implies that ω must be trivial. Since (8.1) is equivalent with
(8.5)
μ
(
K
)
≤
C
(
N
,
p
,
q
)
cap
I
p
-
1
,
q
/
(
q
-
(
p
-
1
)
)
,
N
-
1
(
K
)
for all compact sets K⊂∂ℝ+N≃ℝN-1 (see [21]), we can conclude the following.
Corollary 8.3.
Let 1<p≤N and p-1<q. Let μ in Mb(∂R+N) be nonnegative and suppose u is a nonnegative p-superharmonic solution to -Δpu=2uqH+2μ in RN. If p<N, then
∫
K
u
q
𝑑
x
′
+
μ
(
K
)
≤
C
(
N
,
p
,
q
)
cap
I
p
-
1
,
q
/
(
q
-
(
p
-
1
)
)
,
N
-
1
(
K
)
for all compact sets K⊂∂R+N. If p=N, then u(x′,0)=0 a.e. in ∂R+N and μ≡0.
Since capIp-1,q/(q-(p-1)),N-1≡0 whenever (p-1)qq-(p-1)≥N-1 (see [9]), we can now prove nonexistence for subcritical problems with source.
Proof of Theorem 1.7.
Since every nonnegative local renormalized solution coincides a.e. with a p-superharmonic solution of the same equation, we see that is enough to show that there are no nontrivial nonnegative p-superharmonic solutions of -Δpu=0 in ℝ+N whose trace vanishes a.e. in ∂ℝ+N. But this follows immediately from Remark 7.1. ∎
We finish with a characterization of removable sets for problem (1.5) when μ≡0.
Definition 8.4.
Let 1<p≤N and p-1<q. Given K⊂∂ℝ+N compact, a renormalized solution of (1.6) is a nonnegative function u defined in ℝ+N such that (1)–(3) of Definition 3.6 hold, u∈Lq(Ω) for any closed set Ω⊂∂ℝ+N such that Ω⊂Kc, and there holds
∫
ℝ
+
N
|
∇
u
|
p
-
2
∇
u
⋅
∇
w
d
x
=
∫
∂
ℝ
+
N
∖
K
|
u
|
q
-
1
u
w
𝑑
x
′
for all w∈W1,p(ℝ+N) compactly supported in ℝ+N¯∖K, whose trace belongs to L∞(∂ℝ+N∖K), and satisfying the following condition: there exist k>0, r>N, and functions w±∞∈W1,r(ℝ+N) such that w=w+∞ a.e. in {x∈ℝ+N:u>k} and w=w-∞ a.e. in {x∈ℝ+N:u<-k}.
Definition 8.5.
We say that a compact set K⊂∂ℝ+N is removable for (1.6) if every renormalized solution of (1.6) is a renormalized solution of
(8.6)
{
-
Δ
p
u
=
0
in
ℝ
+
N
,
|
∇
u
|
p
-
2
u
ν
=
u
q
on
∂
ℝ
+
N
.
Proof of Theorem 1.8.
Let u be a renormalized solution to (1.6) and suppose capIp-1,q/(q-(p-1)),N-1(K) is equal to zero. Since q(p-1)q-(p-1)<N-1, we can combine Theorems 5.1.4 and 5.5.1 of [9] to conclude that
cap
1
-
1
/
p
,
p
,
N
-
1
(
K
)
=
0
.
Let u¯ be the extension of u to ℝN by even reflection. Then u¯ is a local renormalized solution to -Δpu¯=2uqℋ in ℝN∖K. By Proposition 2.2, cap1,p,N(K)=0 and by [23, Theorem 4.3.6] this implies that the p-superharmonic representative of u¯ can be extended to ℝN as a nonnegative p-superharmonic function. By Remark 7.3, this p-superharmonic representative coincides cap1,p,N-q.e. with u in ℝ+N¯. Let μ be the Radon measure associated to u¯, i.e., the measure such that -Δpu¯=μ in 𝒟′(ℝN). Let us show that μ=2uqℋ.
Take ϕ∈C0∞(ℝN) nonnegative and let ϕn be such that 0≤ϕn≤ϕ, ϕn∈C0∞(ℝN∖K), and ϕn→ϕ point-wise in ℝN∖K. Note in particular that ϕn→ϕℋ-a.e. Then, by Fatou’s Lemma,
∫
∂
ℝ
+
N
2
u
q
ϕ
𝑑
x
′
≤
lim inf
n
→
∞
∫
ℝ
N
2
u
q
ϕ
n
𝑑
ℋ
=
lim inf
n
→
∞
∫
ℝ
N
|
∇
u
¯
|
p
-
2
∇
u
¯
⋅
∇
ϕ
n
d
x
≤
∫
ℝ
N
ϕ
𝑑
μ
and so we conclude uq∈L1(∂ℝ+N) and μ≥2uqℋ in 𝒟′(ℝN). It follows at once from considering the equations solved by u¯ that in fact μ=2uqℋ in ℝN∖K. Then, setting μK=μ-2uqℋ, we have that u¯ is a p-superharmonic solution of -Δpu¯=2uqℋ+μK in ℝN, where the measure μK is supported in K (and hence bounded). Then, by Corollary 8.3, μK≡0. By [23, Theorem 4.3.4], u¯ is a local renormalized solution to -Δpu¯=2uqℋ in ℝN, and so, by Theorem 5.2, the restriction of u¯ to ℝ+N is a renormalized solution of (8.6).
For the converse, suppose capIp-1,q/(q-(p-1)),N-1(K)>0. We let μ be the capacitary measure of K (see [9, Theorem 2.5.3]) and extend it to ∂ℝ+N by setting μ(A)=μ(A∩K). By [9, Theorem 2.5.5] we see that μ satisfies (8.5) and so, by Theorem 1.6, there exists a renormalized solution of (1.5) with measure ϵμ for some ϵ>0. Since μ is concentrated in K, u is also a solution of (1.6) and thus K is not removable. ∎
Acknowledgements
The author would like to thank Laurent Véron and Marta García-Huidobro. Without their guidance and help this work would not have been possible. The author would also like to extend his gratitude to Institut Henri-Poincaré.
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