1 Introduction
In the paper [1], Benci and Fortunato studied an eigenvalue problem for the Schrödinger operator, coupled with the electromagnetic field. Set in ℝ3, this study lead to the Schrödinger–Maxwell system
(1.1){-12Δv+ψv=ωv,-Δψ=4πv2,
for which the existence of an increasing and divergent sequence of eigenvalues {ωn} was established. Their result was proved using the fact that the solutions of (1.1) are critical points of an indefinite functional, unbounded both from above and below.
In the subsequent paper [3], the related Dirichlet problem with a source term f was studied, that is,
(1.2){-div(M(x)∇u)+Aφ|u|r-2u=f,u∈W01,2(Ω),-div(M(x)∇φ)=|u|r,φ∈W01,2(Ω),
where r>1, A>0, Ω is an open bounded subset of ℝN with N>2,
f belongs to Lm(Ω) with m≥2NN+2,
and M(x) is a symmetric measurable matrix such that
(1.3)(M(x)ξ)ξ≥α|ξ|2,|M(x)|≤β
for almost every x in Ω and every ξ in ℝN, with 0<α≤β.
As in the case of (1.1), solutions of (1.2) are critical points of an indefinite functional, and in [3] it is proved that if
2NN+2≤m≤2NrN+2+4r,r≥N+2N-2,
then there exists a solution (u,φ) in W01,2(Ω)×W01,2(Ω). The fact that φ belongs to W01,2(Ω) is interesting since under the assumption r>N+2N-2, the right-hand side |u|r of the second equation does not belong to the “dual space” L2NN+2(Ω) (recall that 2NN+2=(2*)′).
Hence, the fact that φ belongs to W01,2(Ω) does not follow from it being a solution of the second equation, but from the coupling between u and φ given by the system. In other words, there is a regularizing effect on the solution φ due to the fact it solves a system.
In this paper we improve some existence results of [3], always in the spirit of this regularizing effect. Indeed, we prove that there exist finite energy solutions also when the datum f does not belong to the dual space L2NN+2(Ω), and that one can obtain such solutions by taking data almost in L1(Ω), under the assumption that the exponent r is large enough. Once again, in order to do that, we take advantage of the coupling between the two equations of the system.
Our strategy to prove such a result will be the following. In Section 2 we will prove that, in the case of bounded data f, a solution (u,φ) of (1.2) can be found as a saddle point of the functional
J(z,η)=12∫ΩM(x)∇z∇z-A2r∫ΩM(x)∇η∇η+Ar∫Ωη+|z|r-∫Ωfz,
defined on W01,2(Ω)×W01,2(Ω).
We will then use the solutions found in this case to build an approximating sequence {(un,φn)} of solutions – corresponding to data fn converging to f in Lm(Ω) – which will converge to a solution (u,φ) of (1.2). This and the summability properties of both u and φ, which are the main results of this paper, will be proved in Section 3. In the final section, Section 4, we will prove that the solution we find in Section 3 is still a saddle point of the functional J above, in a suitable sense (thanks to the use of T-minima, introduced in [3]).
2 Data in the dual space
Our first result (which was the starting point in [3]) deals with bounded data f.
In this case, as stated in the introduction, one can find a solution (u,φ) of (1.2) as a saddle point of a suitable functional.
Proposition 2.1.
Let f be in L∞(Ω), and let A>0 and r>1.
Then there exists a weak solution (u,φ) of (1.2).
Furthermore, u and φ belong to L∞(Ω), φ≥0 and (u,φ) is a saddle point of the functional defined on W01,2(Ω)×W01,2(Ω) as
(2.1)J(z,η)={12∫ΩM(x)∇z∇z-A2r∫ΩM(x)∇η∇η+Ar∫Ωη+|z|r-∫Ωfzif ∫Ωη+|z|r<+∞,+∞otherwise.
Proof.
Fix ψ∈W01,2(Ω), and let v=S(ψ) in W01,2(Ω) be the unique minimum of
I1(z)=J(z,ψ).
Note that such a minimum exists since I1 is weakly lower semicontinuous and coercive on W01,2(Ω).
Evidently, v is the unique weak solution of the Euler–Lagrange equation
(2.2)-div(M(x)∇v)+Aψ+|v|r-2v=f,v∈W01,2(Ω).
Observe that, by the classical theory of elliptic equations with discontinuous coefficients and since ψ+≥0, we have
(2.3)∥v∥W01,2(Ω)≤C1∥f∥L∞(Ω),∥v∥L∞(Ω)≤C1∥f∥L∞(Ω).
Consider now the functional
I2(η)=J(v,η).
Since v belongs to L∞(Ω), I2(η) is finite for every η in W01,2(Ω).
Since -I2 is both weakly lower semicontinuous and coercive, there exists a unique maximum ζ=T(v) of I2 on W01,2(Ω).
Since
I2(ζ)≥I2(ζ+),
we have
-A2r∫ΩM(x)∇ζ∇ζ+Ar∫Ωζ+|v|r≥-A2r∫ΩM(x)∇ζ+∇ζ++Ar∫Ω(ζ+)+|v|r,
and it is easy to prove from this inequality that ζ≥0. Observe now that since ζ≥0 is a maximum, we have
-A2r∫ΩM(x)∇ζ∇ζ+Ar∫Ωζ|v|r=-A2r∫ΩM(x)∇ζ∇ζ+Ar∫Ωζ+|v|r
≥-A2r∫ΩM(x)∇ψ∇ψ+Ar∫Ωψ+|v|r
≥-A2r∫ΩM(x)∇ψ∇ψ+Ar∫Ωψ|v|r,
so that ζ is a maximum on W01,2(Ω) of
I3(η)=-A2r∫ΩM(x)∇η∇η+Ar∫Ωη|v|r.
Hence, it is the unique weak solution of the Euler–Lagrange equation
(2.4)-div(M(x)∇ζ)=|v|r,ζ∈W01,2(Ω).
Recalling the estimates
(2.5)∥ζ∥W01,2(Ω)≤C2∥v∥L∞(Ω)r,∥ζ∥L∞(Ω)≤C2∥v∥L∞(Ω)r
and (2.3), we have
∥ζ∥W01,2(Ω)≤C2∥f∥L∞(Ω)r=R,
so that the ball of W01,2(Ω) of radius R is invariant for the map ζ=T(S(ψ)).
We are now going to prove that T∘S satisfies the assumptions of Schauder’s fixed point theorem.
Let {ψn} be a sequence in W01,2(Ω) that is weakly convergent to some ψ, and let vn=S(ψn). Since the sequence {vn} is bounded both in W01,2(Ω) and in L∞(Ω) by (2.3), it follows that (up to subsequences, still denoted by {vn}) it weakly converges to some function v in W01,2(Ω), and strongly converges to the same function in Lq(Ω) for every q>1.
This fact implies that v is the solution of (2.2) with datum ψ, i.e., v=S(ψ). Furthermore, since the sequence {|vn|r} is strongly compact in (say) L2(Ω), classical elliptic estimates imply that the sequence ζn=T(vn) (which is bounded in W01,2(Ω) and in L∞(Ω) by (2.5)) is strongly convergent in W01,2(Ω) to some function ζ, which is the solution of (2.4) with datum v.
That is, ζ=T(v), so that ζ=T(S(ψ)).
We have therefore proved that if {ψn} is bounded in W01,2(Ω), then one can extract from ζn=T(S(ψn)) a subsequence which is strongly convergent in W01,2(Ω), so that T∘S transforms bounded sets of W01,2(Ω) into pre-compact sets of W01,2(Ω).
Furthermore, if ψn is strongly convergent to ψ in W01,2(Ω) and we consider any subsequence {ζnk} of ζn=T(S(ψn)), then a sub-subsequence exists which is strongly convergent in W01,2(Ω) to ζ=T(S(ψ)).
This latter fact follows from the uniqueness results for both (2.2) and (2.4).
Therefore, since the limit does not depend on the subsequence extracted, the whole sequence {T(S(ψn))} converges to ζ=T(S(ψ)).
Thus, we have also proved that T∘S is continuous, and this allows to apply Schauder’s fixed point theorem.
Let φ be the fixed point of T∘S. Observe that since u=S(φ) is a minimum for I1 and φ=T(u)=T(S(φ)) is a maximum of I2, we have
J(u,η)≤J(u,φ)≤J(z,φ) for all z∈W01,2(Ω) and all η∈W01,2(Ω),
and so (u,φ) is a saddle point.
Since φ=T(S(φ)), we have that (u,φ) is a weak solution of
{-div(M(x)∇u)+Aφ|u|r-2u=f,u∈W01,2(Ω),-div(M(x)∇φ)=|u|r,φ∈W01,2(Ω),
with the required regularity properties.
∎
Remark 2.2.
The only place in the proof of the previous theorem where we used that f belongs to L∞(Ω) was to prove (2.3). However, such an estimate holds under the weaker assumption that f belongs to Lm(Ω) with m>N2, thanks to the results of Stampacchia (see [8]). Therefore, the same existence result for (u,φ) can be proved under this weaker assumption.
To conclude this section, we recall the existence result proved in [3], thus completing the “picture” in the case of data in the dual space (which yield finite energy solutions).
Proposition 2.3.
Let f in Lm(Ω) with m≥2NN+2=2*. Then there exists a weak solution (u,φ) of (1.2), with u and φ in W01,2(Ω).
3 Regularizing effect
We now deal with the case of data not in the dual space, proving the main result of this paper. In this case the interplay among the two equations of the system will be crucial in order to obtain estimates. If k≥0, we define the functions
Tk(s)=max(-k,min(s,k)),Gk(s)=s-Tk(s).
Let {fn} be a sequence of L∞(Ω) functions strongly convergent to f in Lm(Ω), m≥1, and such that
(3.1)|fn|≤|f|.
Then, by Theorem 2.1, there exists a solution (un,φn) of the system
(3.2){-div(M(x)∇un)+Aφn|un|r-2un=fn,un∈W01,2(Ω),-div(M(x)∇φn)=|un|r,φn∈W01,2(Ω),
with un and φn in L∞(Ω).
We begin with a result concerning the the positive and negative parts of un, and the truncates of φn; the proof is inspired by the techniques used in [7].
Lemma 3.1.
If (un,φn) is a solution of (3.2), then
(3.3)∫ΩM(x)∇T1(φn)∇w≥∫{0≤φn≤1}|un|rw
for every w in W01,2(Ω), w≥0.
Proof.
Let Hε(s)=1-1εTε(G1(s+)), and choose Hε(φn)w as test function in the second equation of (3.2), with w in W01,2(Ω), w≥0. We have
-1ε∫{1≤φn≤1+ε}M(x)∇φn∇φnw+∫ΩM(x)∇φn∇wHε(φn)=∫Ω|un|rwHε(φn).
Since the first term is negative by (1.3), we can drop it, obtaining
∫ΩM(x)∇φn∇wHε(φn)≥∫Ω|un|rwHε(φn).
Letting ε tend to zero, we obtain (3.3).
∎
Our next result deals with a priori estimates on {un}.
Lemma 3.2.
Let m>1 and k≥0. Then there exists C0>0, independent of n and k, such that
(3.4)∫Ω|Gk(un)|m(r-1)≤C0∫{|un|≥k}|f|m+C0meas({|un|≥k})
and
(3.5)(∫Ω|Gk(un)|m**)1m**≤C0(∫{|un|≥k}|f|m)1m,
In particular (choosing k=0), {|un|ρ} is bounded in L1(Ω) with ρ=max(m(r-1),m**).
Proof.
Let γ>-1.
First, we work with the first inequality. Let ε>0, and choose (|Gk(un)|+ε)γGk(un) as test function. Note that since γ may be negative, we need to “add” the term with ε since the gradient of |Gk(un)|γGk(un) may be not defined when Gk(un)=0, even though |Gk(un)|γGk(un) is well defined since γ+1>0.
After using (1.3) and (3.1), and dropping the positive term involving the principal part, we obtain
A∫Ωφn|un|r-1(|Gk(un)|+ε)γ|Gk(un)|≤∫Ω|f|(|Gk(un)|+ε)γ|Gk(un)|,
which implies, by letting ε tend to zero (recall that every un is a bounded function), that
A∫Ωφn|un|r-1|Gk(un)|γ+1≤∫Ω|f||Gk(un)|γ+1.
Therefore, since |Gk(un)|≤|un| and r>1, we have
A∫{φn≥1}|Gk(un)|γ+r≤A∫Ωφn|Gk(un)|γ+r
=A∫Ωφn|Gk(un)|r-1|Gk(un)|γ+1
≤A∫Ωφn|un|r-1|Gk(un)|γ+1
(3.6)≤∫Ω|f||Gk(un)|γ+1.
Now we work with the second equation in two different ways, according to the value of γ.
Suppose that γ>1. Choose w=|Gk(un)|γ as test function in (3.3) to obtain
(3.7)γ∫ΩM(x)∇T1(φn)∇Gk(un)|Gk(un)|γ-2Gk(un)≥∫{0≤φn≤1}|un|r|Gk(un)|γ≥∫{0≤φn≤1}|Gk(un)|γ+r.
On the other hand, choosing γT1(φn)(|Gk(un)|+ε)γ-2Gk(un) as test function in the first equation of (3.2), by dropping the terms
γ∫ΩM(x)∇un∇[(|Gk(un)|+ε)γ-2Gk(un)]T1(φn)
and
Aγ∫Ωφn|un|r-2T1(φn)(|Gk(un)|+ε)γ-2unGk(un),
which are positive, we get
γ∫ΩM(x)∇Gk(un)∇T1(φn)(|Gk(un)|+ε)γ-2Gk(un)≤γ∫Ω|f|(|Gk(un)|+ε)γ-1.
Letting ε tend to zero, we obtain
γ∫ΩM(x)∇Gk(un)∇T1(φn)|Gk(un)|γ-2Gk(un)≤γ∫Ω|f||Gk(un)|γ-1,
which, together with (3.7) and the fact that |Gk(un)|γ-1≤|Gk(un)|γ+1+1 (since γ>1), implies that
(3.8)∫{0≤φn≤1}|Gk(un)|γ+r≤γ∫Ω|f||Gk(un)|γ-1≤γ∫Ω|f||Gk(un)|γ+1+γ∫{|un|≥k}|f|.
Therefore, (3.6) and (3.8) give
∫{φn≥1}|Gk(un)|γ+r+∫{0≤φn≤1}|Gk(un)|γ+r≤(1A+γ)∫Ω|f||Gk(un)|γ+1+γ∫{|un|≥k}|f|,
that is,
(3.9)∫Ω|Gk(un)|γ+r≤(1A+γ)∫Ω|f||Gk(un)|γ+1+γ∫{|un|≥k}|f| for all γ>1.
Now we study the case -1<γ≤1, and choose w=|Gk(un)| as test function in (3.3) to obtain
(3.10)∫ΩM(x)∇T1(φn)∇Gk(un)sgn(Gk(un))≥∫{0≤φn≤1}|un|r|Gk(un)|≥∫{0≤φn≤1}|Gk(un)|r+1.
Choosing T1(φn)1εTε(Gk(un)) in the first equation of (3.2), and dropping two positive terms, we get
∫ΩM(x)∇Gk(un)∇T1(φn)1εTε(Gk(un))≤∫{|un|≥k}|f|,
so that, by letting ε tend to zero, we get
∫ΩM(x)∇Gk(un)∇T1(φn)sgn(Gk(un))≤∫{|un|≥k}|f|,
which, together with (3.10), yields
(3.11)∫{0≤φn≤1}|Gk(un)|r+1≤∫{|un|≥k}|f|.
Since γ≤1, we have
∫{0≤φn≤1}|Gk(un)|γ+r≤∫{0≤φn≤1}|Gk(un)|r+1+meas({|un|≥k})≤∫{|un|≥k}|f|+meas({|un|≥k}),
which, together with (3.6), implies that
(3.12)∫Ω|Gk(un)|γ+r≤1A∫Ω|f||Gk(un)|γ+1+∫{|un|≥k}|f|+meas({|un|≥k}).
Summing up the results of (3.9) and (3.12), for every γ>-1, we obtain
∫Ω|Gk(un)|γ+r≤C1∫Ω|f||Gk(un)|γ+1+C1∫{|un|≥k}|f|+C1meas({|un|≥k}).
Observe now that we have
∫Ω|f||Gk(un)|γ+1=∫{|f|≤δ|Gk(un)|r-1}|f||Gk(un)|γ+1+∫{δ|Gk(un)|r-1<|f|}|f||Gk(un)|γ+1
≤δ∫Ω|Gk(un)|γ+r+Cδ∫{|un|≥k}|f|γ+rr-1.
We now choose γ such that γ+rr-1=m. This implies that γ>-1, since m>1, and that γ+r=m(r-1). Thus, choosing δ such that C1δ=12, we have
∫Ω|Gk(un)|γ+r≤12∫Ω|Gk(un)|γ+r+C2∫{|un|≥k}|f|m+C2∫{|un|≥k}|f|+C2meas({|un|≥k}),
so that
∫Ω|Gk(un)|m(r-1)≤C3∫{|un|≥k}|f|m+C3meas({|un|≥k}),
which is (3.4).
To obtain (3.5), suppose that γ>12, and choose (|Gk(un)|+ε)2γ-2Gk(un) as test function in the first equation of (3.2) with ε>0. Since
∇[(|Gk(un)|+ε)2γ-2Gk(un)]=∇Gk(un)(|Gk(un)|+ε)2γ-3[(2γ-1)|Gk(un)|+ε]
and
(2γ-1)|Gk(un)|+ε≥min(2γ-1,1)(|Gk(un)|+ε),
we have, by dropping a positive term and using (1.3), that
αmin(2γ-1,1)∫Ω|∇Gk(un)|2(|Gk(un)|+ε)2γ-2≤∫Ω|f|(|Gk(un)|+ε)2γ-1.
Using the Sobolev embedding on the left, Hölder’s inequality on the right, and then letting ε tend to zero, we have
α𝒮min(2γ-1,1)γ2(∫Ω|Gk(un)|2*γ)22*≤∫Ω|f||Gk(un)|2γ-1
=∫{|un|≥k}|f||Gk(un)|2γ-1
≤(∫{|un|≥k}|f|m)1m(∫Ω|Gk(un)|(2γ-1)m′)1m′.
Choosing γ=m**2*, and simplifying equal terms, yields
(∫Ω|Gk(un)|m**)1m**≤C0(∫{|un|≥k}|f|m)1m,
as desired.
∎
Remark 3.3.
If m=1, the estimate proved in [4] implies that the sequence {un} is bounded in Ls(Ω) for every s<NN-2. On the other hand, choosing Tε(un)/ε as test function in the first equation of (3.2), by observing that unTε(un)=|un||Tε(un)| and dropping a positive term, we have
A∫{φn≥1}|un|r-1|Tε(un)|ε≤A∫Ωφn|un|r-1|Tε(un)|ε≤∫ΩfnTε(un)ε≤∥f∥L1(Ω).
Letting ε tend to zero, we obtain
A∫{φn≥1}|un|r-1≤∥f∥L1(Ω).
On the other hand, (3.11) with k=0 implies that
∫{0≤φn≤1}|un|r-1≤∫{0≤φn≤1}|un|r+1+meas(Ω)≤∥f∥L1(Ω)+meas(Ω),
so that we have
∫Ω|un|r-1≤(1+1A)∥f∥L1(Ω)+meas(Ω).
Thus, since r-1≥NN-2 if r≥2N-1N-2, we have that
{un} is bounded in Ls(Ω) for all s<NN-2 if 1<r<2N-1N-2
and bounded in Lr-1(Ω) if r≥2N-1N-2.
Now we recall that for the single equation
-div(M(x)∇w)+a(x)w|w|r-2=f,a(x)≥a0>0,
it has been proved in [6] that w belongs to W01,2(Ω) if f belongs to Lm(Ω) with m≥r′.
In our case a(x) is φ(x) and we only know that φ(x)≥0.
Nevertheless, we are able to prove (in the next result, the main of this paper) that the solution of the first equation belongs to W01,2(Ω), under the same assumption m≥r′, using the fact that we are dealing with a system.
Theorem 3.4.
Let r>2*, and let f in Lm(Ω), with
rr-1=r′≤m<2NN+2.
Then there exists a weak solution (u,φ) of system (1.2), with u and
φ
in W01,2(Ω).
Let 1<r<2*, and let f in Lm(Ω), with
max(NrN+2r,1)<m<2NN+2.
Then there exists a weak solution (u,φ) of system (1.2) with u in W01,m*(Ω), and
φ
in W01,q(Ω) with
q={2if 2NrN+2+4r≤m<2NN+2,NmNr-2mr-mif max(NrN+2r,1)<m<2NrN+2+4r.
Remark 3.5.
If r=2*, both cases of the previous theorem “collapse” to m=2NN+2, and in this case the existence result is given by Proposition 2.3.
Proof.
We begin with case (A): r>2*.
Thanks to Lemma 3.2, the sequence {un} is bounded in Lm(r-1)(Ω) (note that we have m(r-1)>r-1>N+2N-2>1 for every m>1), and in Lm**(Ω). Since
m(r-1)≥m**⇔m≤N2r-2r-1
and
N2r-2r-1>2NN+2 for every r>2*,
we have m(r-1)>m** for every r>2* and every 1<m<2NN+2, so that the better estimate on {un} is the boundedness in Lm(r-1)(Ω), m>1.
Thus, {|un|r} is bounded in Lmr′(Ω).
In order to continue, we have to make a further restriction on m.
Namely, m≥r′, since we need the sequence {|un|r} to be bounded at least in L1(Ω), in order to obtain estimates on {φn} using the second equation of (3.2).
Therefore, from now on, we are working with the assumption
r>2*,r′≤m<2NN+2.
We choose now un as test function in the first equation of (3.2). Using (1.3), we have
α∫Ω|∇un|2+A∫Ωφn|un|r≤∫Ωfnun≤∥f∥Lm(Ω)∥un∥Lm′(Ω).
Here we note that the assumption r′≤m implies that m′≤m(r-1),
therefore {un} is bounded in W01,2(Ω) and
∫Ωφn|un|r≤C1.
Choosing φn as test function in the second equation of (3.2), using (1.3) and the last estimate, yields
α∫Ω|∇φn|2≤∫Ωφn|un|r≤C1,
so that also {φn} is bounded in W01,2(Ω).
Thus, up to subsequences, (un,φn) converges to (u,φ) weakly in (W01,2(Ω))2, strongly in (Lp(Ω))2 for every p<2*, and almost everywhere in Ω.
In order to prove that (u,φ)
is a solution of the system, we have to pass to the limit in the two nonlinear terms.
However, the strong convergence in Lp(Ω) with p<2*<r, is not enough to pass to the limit, so that we will have to use again Lemma 3.2.
Indeed, since m(r-1)≥r by the assumption m≥r′, inequality (3.4) implies that
(3.13)∫Ω|Gk(un)|r≤C2(∫{|un|≥k}|f|m+meas({|un|≥k}))rm(r-1).
Therefore, if E is a measurable subset of Ω, then we have
∫E|un|r≤2r-1∫E|Tk(un)|r+2r-1∫E|Gk(un)|r
(3.14)≤2r-1krmeas(E)+2r-1∫Ω|Gk(un)|r.
Let ε>0, and fix k0>0 so that
2r-1∫Ω|Gk0(un)|r≤ε2 for all n∈ℕ.
This can be done (using (3.13)) since f belongs to Lm(Ω), and the measure of {|un|≥k} is uniformly (in n) small if k is large by the a priori estimates on {un}.
Once k0 is fixed, choose meas(E) small enough so that
2r-1k0rmeas(E)<ε2 for all n∈ℕ.
Thus, by (3.14), the sequence {|un|r} is equiintegrable, and by Vitali’s theorem we have
limn→+∞∫Ω|un|r=∫Ω|u|r.
This is enough to pass to the limit in the second equation of (3.2), with test functions in W01,2(Ω)∩L∞(Ω). Choose now 1εTε(Gk(un)) as test function in the first equation of (3.2). By dropping a positive term, and then letting ε tend to zero, we get
(3.15)∫{|un|≥k}φn|un|r-1≤1A∫{|un|≥k}|fn|≤1A∫{|un|≥k}|f|.
Thus, if E is a measurable subset of Ω, then
∫Eφn|un|r-1≤∫Eφn|Tk(un)|r-1+∫E∩{|un|≥k}φn|un|r-1≤kr-1∫Eφn+C3∫{|un|≥k}|f|.
As before, for fixed ε>0, we use the estimates on un, and the fact that f belongs to L1(Ω), to choose k0>0 such that
C3∫{|un|≥k}|f|<ε2 for all n∈ℕ,
and then use the fact that φn is strongly compact in L1(Ω) (by Rellich’s theorem) to choose meas(E) small enough so that
k0r-1∫Eφn<ε2 for all n∈ℕ.
Thus, the sequence {φn|un|r-1} is equiintegrable, so that, by Vitaly’s theorem,
limn→+∞∫Ωφn|un|r-1=∫Ωφ|u|r-1.
Now this convergence is enough to pass to the limit in the first equation of (3.2), with test functions in W01,2(Ω)∩L∞(Ω).
We turn now to the case (B): 1<r<2*.
Note that if r<2*, then
N2r-2r-1<NrN+2r<2NN+2<r′.
Using Lemma 3.2, we have that {|un|ρ} is bounded in L1(Ω), with
ρ={m**if m≥N2r-2r-1, m>1,m(r-1)if 1<m<N2r-2r-1.
In order to have that {|un|r} is bounded at least in L1(Ω), we need ρ≥r, so that
{m**r≥1if m≥N2r-2r-1, m>1,m(r-1)r≥1if 1<m<N2r-2r-1, that is, {NrN+2r≤m<2NN+2,m>1,r′≤m<N2r-2r-1.
However, since r<2*, we have N2r-2r-1<r′,
so that the second inequality above is never satisfied.
Thus, the
range of possible values of r and m becomes
(3.16)1<r<2*,NrN+2r≤m<2NN+2,m>1.
Since the summability of un is the one obtained using only the principal part of the operator in the first equation and not the lower order term, one cannot expect boundedness in W01,2(Ω) for the sequence {un}. In other words, the regularizing effect of the second equation on un is lost in this case, and one only has the estimate proved in [5]. More precisely,
{un} is bounded in W01,m*(Ω).
As for φn, we can use the second equation of (3.2), and the fact that {|un|r} is bounded in Ls(Ω), with s=m**r≥1 by (3.16).
Again, by the theory of elliptic equations, we have the following cases (see [4, 5]):
{φn} is bounded in W01,2(Ω) if s≥2NN+2,
{φn} is bounded in W01,s*(Ω) if 1<s<2NN+2,
{φn} is bounded in W01,q(Ω) for all q<NN-1 if s=1.
Note that
2NrN+2+4r≤m<2NN+2⟹s≥2NN+2,
max(NrN+2r,1)<m<2rNN+4r+2⟹1<s<2NN+2,
and s=1 if m=NrN+2r and m>1.
Therefore, the following hold:
{φn} is bounded in W01,2(Ω) if 2NrN+2+4r≤m<2NN+2,
{φn} is bounded in W01,s*(Ω) if max(NrN+2r,1)<m<2NrN+2+4r,
{φn} is bounded in W01,q(Ω) for all q<NN-1 if m=NrN+2r, m>1.
Since
s*=(m**r)*=NmNr-2mr-m,
we have the desired estimates on φn.
In order to pass to the limit in the approximate equations, we can follow the same steps as in the case r>2*, using (3.5) for the second equation of (3.2), and (3.15) (which still holds) for the nonlinear term of the first one.
∎
Remark 3.6.
Theorem 3.4 does not deal with the case m=1 and 1<r<NN-2.
However, following the proof of (B) above, we can prove the following:
the sequence {un} is bounded in W01,q(Ω) for every q<NN-1,
{φn} is bounded in W01,2(Ω) if 1<r<N+22(N-2),
{φn} is bounded in W01,ρ(Ω) if N+22(N-2)≤r<NN-2
with 1≤ρ<NNr-2r-1.
Note that the first case above is empty if N≥6 since r>1.
4 Saddle points
In Proposition 2.1, we have proved that if f is bounded, then there exists a saddle point (u,φ) of the functional J defined in (2.1).
Thus,
the approximating solution (un,φn) of (3.2) can be found as a saddle point of the functional Jn defined as
(4.1)Jn(v,ψ)=12∫ΩM(x)∇v∇v-A2r∫ΩM(x)∇ψ∇ψ+Ar∫Ωψ+|v|r-∫Ωfnv,
in the sense that
Jn(un,ψ)≤Jn(un,φn)≤Jn(v,φn) for all v∈W01,2(Ω) and all ψ∈W01,2(Ω).
In this section, we study how the convergences on (un,φn), proved in the previous section, can be used in order to give a meaning to the concept of “saddle point of J” if the function f does not belong to L∞(Ω) (actually, if it does not belong to L2*(Ω)). Note that in this case the functional J is not well defined, since the are two terms (with opposite signs) which can possibly be unbounded.
Recalling that (un,φn) is a saddle point for Jn(v,ψ) and that φn is positive, we have
12∫ΩM(x)∇un∇un-A2r∫ΩM(x)∇ψ∇ψ+Ar∫Ωψ+|un|r-∫Ωfnun
≤12∫ΩM(x)∇un∇un-A2r∫ΩM(x)∇φn∇φn+Ar∫Ωφn|un|r-∫Ωfnun
≤12∫ΩM(x)∇v∇v-A2r∫ΩM(x)∇φn∇φn+A2r∫Ωφn|v|r-∫Ωfnv
for every v and ψ in W01,2(Ω), with v such that
∫Ωφn|v|r<+∞.
Splitting the above inequalities, simplifying equal terms, and rearranging them, we get
12∫ΩM(x)∇φn∇φn-∫Ωφn|un|r≤12∫ΩM(x)∇ψ∇ψ-∫Ωψ+|un|r
and
12∫ΩM(x)∇un∇un+Ar∫Ωφn|un|r-∫Ωfnun≤12∫ΩM(x)∇v∇v+A2r∫Ωφn|v|r-∫Ωfnv.
Now we follow [3] and choose ψ=φn-Tk(φn-η) with η in W01,2(Ω)∩L∞(Ω) and η≥0 in the first inequality, and we obtain
12∫ΩM(x)∇φn∇φn+∫Ω[φn-Tk(φn-η)]+|un|r
≤12∫ΩM(x)∇[φn-Tk(φn-η)]∇[φn-Tk(φn-η)]+∫Ωφn|un|r.
We observe now that ∇[φn-Tk(φn-η)] is ∇φn where |φn-η|>k, and ∇η where |φn-η|≤k, and that, since η≥0, [φn-Tk(φn-η)]+=φn-Tk(φn-η). Therefore, simplifying again equal terms, we have
12∫{|φn-η|≤k}M(x)∇φn∇φn≤12∫{|φn-η|≤k}M(x)∇η∇η+∫ΩTk(φn-η)|un|r.
Recall now that, under any assumption on f and r, we proved that un converges to u strongly in Lr(Ω), so that one can pass to the limit in the last term. Furthermore, using Lebesgue’s theorem, one can easily pass to the limit in the second term. As for the first one, we remark that on the set {|φn-η|≤k} one has |φn|≤h, where h=k+∥η∥L∞(Ω). Therefore, one has
∫{|φn-η|≤k}M(x)∇φn∇φn=∫{|φn-η|≤k}M(x)∇Th(φn)∇Th(φn).
Recalling that Tk(φn) weakly converges to Tk(φ) in W01,2(Ω) (since {φn} or {Th(φn)} is bounded in W01,2(Ω), depending on the assumptions on f and r), we can use weak lower semicontinuity to have that
12∫{|φ-η|≤k}M(x)∇φ∇φ≤12∫{|φ-η|≤k}M(x)∇η∇η+∫ΩTk(φ-η)|u|r
for every η in W01,2(Ω)∩L∞(Ω), η≥0.
Using the inequality for un, and choosing v=un-Tk(un-w) with w in W01,2(Ω)∩L∞(Ω), as before, we obtain
12∫{|un-w|≤k}M(x)∇un∇un+Ar∫Ωφn[|un|r-|un-Tk(un-w)|r]≤12∫{|un-w|≤k}M(x)∇w∇w+∫ΩfnTk(un-w).
Observe now that since
||s|r-|s-t|r|≤r(|s|+|t|)r-1|t|,
we have
||un|r-|un-Tk(un-w)|r|≤r(|un|+|Tk(un-w)|)r-1|Tk(un-w)|≤rk(|un|+k)|r-1.
Recall now that we have proved, under any assumption on f and r, that φn|un|r-1 strongly converges in L1(Ω) to φ|u|r-1. Therefore, Vitaly’s theorem implies that
limn→+∞∫Ωφn[|un|r-|un-Tk(un-w)|r]=∫Ωφ[|u|r-|u-Tk(u-w)|r].
Hence, recalling that Tk(un) weakly converges to TK(u) in W01,2(Ω), and observing, as before, that
∫{|un-w|≤k}M(x)∇un∇un=∫{|un-w|≤k}M(x)∇Th(un)∇Th(un),
where h=k+∥w∥L∞(Ω), we have
12∫{|u-w|≤k}M(x)∇u∇u+Ar∫Ωφ[|u|r-|u-Tk(u-w)|r]≤12∫{|u-w|≤k}M(x)∇w∇w+∫ΩfTk(u-w)
for every w in W01,2(Ω)∩L∞(Ω).
We have therefore proved the following result. Note that (4.2) and (4.3) below state that both u and φ are T-minima of suitable functionals (see [2] for the definition of T-minimum of a functional).
Theorem 4.1.
Under the same assumptions on f and r made in Theorems 2.3 and 3.4, if (un,φn) is a saddle point of the functional Jn(v,ψ) defined in (4.1), and if (u,φ) is the solution of (1.2) obtained as limit of (un,φn),
then we have
(4.2)12∫{|φ-η|≤k}M(x)∇φ∇φ≤12∫{|φ-η|≤k}M(x)∇η∇η+∫ΩTk(φ-η)|u|r
for every η in W01,2(Ω)∩L∞(Ω), η≥0, and
(4.3)12∫{|u-w|≤k}M(x)∇u∇u+Ar∫Ωφ[|u|r-|u-Tk(u-w)|r]≤12∫{|u-w|≤k}M(x)∇w∇w+∫ΩfTk(u-w)
for every w in W01,2(Ω)∩L∞(Ω).
