1 Introduction and Main Results
Let Ω⊂ℝN (N≥1) be a bounded domain with smooth boundary ∂Ω. In this paper, we consider nonnegative solutions of the problem
(PB)
{
-
Δ
u
=
λ
a
(
x
)
f
(
u
)
+
b
(
x
)
g
(
u
)
in
Ω
,
ℬ
u
=
0
on
∂
Ω
,
where
Δ is the usual Laplacian in ℝN;
ℬ
u
:
=
u
(Dirichlet) or ℬu:=∂u∂𝐧 (Neumann), where 𝐧 is the outward unit normal to ∂Ω;
λ
∈
ℝ
is a bifurcation parameter;
a
,
b
∈
C
(
Ω
¯
)
are such that a changes sign in Ω and b(x0)>0 for some x0∈Ω;
f
,
g
:
[
0
,
∞
)
→
ℝ
are continuous functions with f(0)=g(0)=0.
It follows that (PB) possesses the trivial line (λ,0) of zero solutions. The prototype of f,g to be considered in this paper is
(1.1)
f
(
s
)
=
s
q
,
g
(
s
)
=
s
p
,
with
0
<
q
<
1
<
p
,
so that the nonlinearity λf(s)+g(s) has a concave-convex nature.
More precisely, we assume that f∈C1((0,∞)) with f(s)>0 for s>0 and g∈C1([0,∞)) satisfy
(1.2)
lim
s
→
0
+
f
(
s
)
s
=
∞
,
(1.3)
lim
s
→
0
+
g
(
s
)
s
=
0
,
(1.4)
lim
s
→
∞
f
(
s
)
s
=
0
,
(1.5)
lim
s
→
∞
g
(
s
)
s
=
∞
.
Let r>N. A function u∈W2,r(Ω) (and consequently, u∈C1(Ω¯)) is said to be a nonnegative solution of (PB) if u≥0 in Ω, u satisfies the equation pointwisely a.e. in Ω, and ℬu=0 on ∂Ω. If, in addition, u satisfies
{
u
>
0
in
Ω
and
∂
u
∂
𝐧
<
0
on
∂
Ω
if
ℬ
u
=
u
,
u
>
0
on
Ω
¯
if
ℬ
u
=
∂
u
∂
𝐧
,
then we write u≫0. In this case u lies in the interior of the positive cone {u∈V:u≥0}, where
V
:
=
{
C
0
1
(
Ω
¯
)
:
=
{
u
∈
C
1
(
Ω
¯
)
:
u
=
0
on
∂
Ω
}
if
ℬ
u
=
u
,
C
1
(
Ω
¯
)
if
ℬ
u
=
∂
u
∂
𝐧
.
A nonnegative solution u of (PB) is called positive if u≫0.
Our first goal is to establish, under certain conditions on a and b, the existence of loop type subcontinua{(λ,u)} (i.e., nonempty, closed and connected subsets in ℝ×V) composed by (0,0) and nontrivial nonnegative solutions (λ,u) of (PB). We shall prove the existence of a loop type subcontinuum 𝒞0 such that
(1.6)
𝒞
0
∩
{
(
λ
,
0
)
:
λ
∈
ℝ
}
=
{
(
0
,
0
)
}
.
It should be emphasized that, in general, one can not deduce that nontrivial nonnegative solutions of (PB) satisfy u≫0, since the strong maximum principle does not apply. This is due to the fact that a(x)f(⋅) does not satisfy the slope condition [1, p. 623], see Remark 2.1 (ii) below. As a matter of fact, (PB) may have solutions u satisfying u>0 in Ω but not u≫0 (for concrete examples one may argue as in the proof of [15, Proposition 2.9] after a slight modification). In view of this difficulty, our second purpose is to show that nontrivial solutions lying on 𝒞0 satisfy u≫0.
Bounded subcontinua of positive solutions for indefinite superlinear equations of the form
-
Δ
u
=
λ
a
(
x
)
u
+
b
(
x
)
u
p
in
Ω
,
with Ω bounded (under different boundary conditions) or Ω=ℝN, have been studied by several authors, see e.g. [4, 5, 8, 6, 21, 7, 19, 18]. According to [6, 7, 19], a bounded subcontinuum linking two different points on (λ,0) is called a mushroom, one that meets a single point on (λ,0) is called a loop, and one that does not touch (λ,0) is called an isola. Cingolani and Gámez studied both the Dirichlet condition case and the case Ω=ℝN, proving the existence of mushrooms [8, Theorems 4.4 and 5.5]. Cano-Casanova considered a mixed boundary condition (with a second order uniformly strongly elliptic operator), and proved the existence of a mushroom [5, Theorem 1.4]. López-Gómez and Molina-Meyer dealt with the Dirichlet condition, and established existence results for a mushroom, a loop and an isola in three cases, respectively [19, Theorems 3.1, 5.1 and 5.2]. In the case of Neumann boundary conditions, Brown proved the existence of a mushroom and a loop in two situations, respectively [4, Sections 2 and 5]. Finally, we refer to [26, Section 3] for the existence of a mushroom of positive solutions for a semilinear elliptic problem with a logistic nonlinearity and an indefinite weight, coupled with a nonlinear boundary condition. Let us emphasize that all the previous works hold in the regular case, i.e., when the nonlinearity considered is C1 at u=0, so that the general theory on local and global bifurcation from simple eigenvalues can be directly applied.
Regarding existence results for positive solutions of concave-convex problems, a large number of works have been devoted to (PB) in the “definite case” (i.e. with a≥0, a≢0) since the classical work of Ambrosetti, Brezis and Cerami [3], which treats the model case (1.1) with a=b≡1 and p≤N+2N-2 under the Dirichlet boundary condition. In [3] it is proved that (PB) has two positive solutions for λ>0 sufficiently small. This result was extended by De Figueiredo, Gossez and Ubilla [11] to the non-powerlike case, with a≥0. In addition, in [10], the authors allowed a to change sign and proved the existence of two nontrivial nonnegative solutions of (PB) for λ>0 small. We refer to [23] for a discussion on concave-convex problems under the Neumann boundary condition.
To the best of our knowledge, besides [15, 23] there are no works providing the existence of solutions that are positive in Ω for indefinite concave-convex problems (i.e., with a changing sign). In [15, 16, 14], we first established a positivity property for (PB) in the powerlike and concave case, i.e. with f(s)=sq and b≡0. Thanks to these results, we obtained a positivity result for (PB) with f(s)=sq and b≡1 (see [15, Section 4]). Finally, let us mention that in the model case (1.1), the existence of a loop type subcontinuum of nonnegative solutions for the Neumann case was obtained by means of a bifurcation approach in [23]. Furthermore, the asymptotic profile of nonnegative solutions as λ→0+ enables one to deduce in some cases their positivity for λ>0 small, cf. [23, Corollary 1.3].
For our first purpose, we assume that there exist two balls B,B′⋐Ω and constants a0,a0′,b0,b0′>0 such that
(1.7)
{
a
(
x
)
≥
a
0
and
b
(
x
)
≥
b
0
in
B
,
-
a
(
x
)
≥
a
0
′
and
b
(
x
)
≥
b
0
′
in
B
′
.
Let ψ∈C(Ω¯) be such that
Ω
+
ψ
:
=
{
x
∈
Ω
:
ψ
(
x
)
>
0
}
≠
∅
.
Then, we introduce the condition
(Hψ)
Ω
+
ψ
consists of a finite number of connected components of
Ω
.
We shall assume this condition for ψ=a and ψ=-a.
Motivated by the model case (1.1), we assume that
(1.8)
lim
s
→
0
+
s
1
-
q
f
′
(
s
)
=
:
f
0
∈
(
0
,
∞
)
for some
q
∈
(
0
,
1
)
.
We will see that under this condition f behaves like f0qsq when s→0+, and satisfies the slope condition, see Remark 2.1. In addition, the following strong concavity (resp. convexity) condition on f (resp. g) shall be used:
(1.9)
(
f
(
s
)
s
q
)
′
≤
0
for
s
>
0
,
(1.10)
(
g
(
s
)
s
)
′
>
0
for
s
>
0
,
where q∈(0,1) is given by (1.8). We introduce now the Gidas–Spruck condition [12, Theorem 1.1], which is stronger than (1.5),
(1.11)
0
<
lim
s
→
∞
g
(
s
)
s
p
<
∞
for some
p
>
1
,
where
p
<
N
+
2
N
-
2
if
N
>
2
.
Finally, we shall use condition (Hb), which will be precisely stated in Remark 2.3 and goes back to Amann and López-Gómez [2].
For our second purpose, we focus on the case f(s)=sq, q∈(0,1). In association with the sublinear problem
(1.12)
{
-
Δ
u
=
a
(
x
)
u
q
in
Ω
,
ℬ
u
=
0
on
∂
Ω
,
we introduce the set
(1.13)
𝒜
ℬ
a
:
=
{
q
∈
(
0
,
1
)
:
u
≫
0
for any nontrivial nonnegative solution
u
of (1.12)
}
.
We know [15, Corollary 1.5 and Theorem 1.9] that under (Hψ) with ψ=a, there exists qa∈[0,1) such that 𝒜ℬa=(qa,1), assuming additionally ∫Ωa<0 if ℬu=∂u∂𝐧. Let us point out that the condition ∫Ωa<0 is necessary and sufficient for the existence of a positive solution of (1.12) with ℬu=∂u∂𝐧, for some q∈(0,1), see [16, Corollary 1.3].
We are now in position to state our main results. First, we deal with the Dirichlet problem.
Theorem 1.1.
Under Bu=u, we assume (1.3), (1.4), (1.7), (1.8), (1.11) and (Hψ) with ψ=±a. In addition, suppose either
b
>
0
on
Ω
¯
, and
p
<
N
+
2
N
-
2
if
N
>
2
, or
(
1.9
), (
1.10
) and
(Hb).
Then, the following two assertions hold:
(
(PB)
) admits a loop type
subcontinuum
𝒞
0
(i.e., a nonempty, closed and connected subset in
ℝ
×
C
0
1
(
Ω
¯
)
) of nonnegative solutions which satisfies (
1.6
). Moreover, we have the following properties, see Figure
1
(i):
(
0
,
u
0
)
∈
𝒞
0
for some positive solution
u
0
of (
(PB)
) with
λ
=
0
.
There exists
δ
>
0
such that
𝒞
0
does not contain any positive solution
u
of (
(PB)
) with
λ
=
0
satisfying
∥
u
∥
C
(
Ω
¯
)
≤
δ
.
𝒞
0
contains closed connected sets
𝒞
0
±
such that
{
(
0
,
0
)
}
⊊
𝒞
0
±
, and if
(
λ
,
u
)
∈
𝒞
0
±
∖
{
(
0
,
0
)
}
, then
λ
≷
0
, i.e.,
𝒞
0
bifurcates both subcritically and supercritically at
(
0
,
0
)
.
Let
f
(
s
)
=
s
q
, q∈(0,1). Assume that b≥0, and additionally that
(1.14)
g
(
s
)
≥
0
𝑓𝑜𝑟
s
>
0
when condition (a) holds. If q∈𝒜ℬa∩𝒜ℬ-a, then u≫0 for any (λ,u)∈𝒞0∖{(0,0)}. In particular, the component of nontrivial nonnegative solutions of ((PB)) including 𝒞0∖{(0,0)} is bounded.
Figure 1
The loop type subcontinua 𝒞0 and 𝒞∗.
Next, we consider the Neumann problem under the condition
(1.15)
∫
Ω
b
<
0
.
In this case, we shall obtain a loop type subcontinuum of nonnegative solutions with the same nature the Dirichlet case admits, as in Figure 1 (i) (for the case ∫Ωb≥0 we refer to Remark 1.3 below). To this end, we need the following decay and positivity condition for g, which is stronger than (1.3):
(1.16)
lim
s
→
0
+
g
(
s
)
s
σ
=
:
g
0
∈
(
0
,
∞
)
for some
σ
>
1
,
where
σ
<
2
N
N
-
2
if
N
>
2
.
Moreover, we are able to discuss the positivity of (nontrivial) nonnegative solutions for (PB) with (1.1), assuming
(1.17)
{
p
<
N
+
1
N
-
1
if
N
>
2
,
a
∈
C
α
(
Ω
¯
)
for some
α
∈
(
0
,
1
)
,
∫
Ω
a
<
0
,
b
≡
1
.
It is known [25, Theorem 1] that if Ω+a is connected, then (PB) possesses a loop type subcontinuum 𝒞∗ in ℝ×C(Ω¯) of nonnegative solutions which satisfies (1.6). Furthermore, we have the following properties, see Figure 1 (ii):
{
λ
:
(
λ
,
u
)
∈
𝒞
∗
∖
{
(
0
,
0
)
}
}
=
(
0
,
Λ
∗
]
for some Λ∗>0.
𝒞
∗
possesses at least two nontrivial nonnegative solutions for λ>0 small enough.
Now, we state our main results for the Neumann problem, which are given in a similar way as in Theorem 1.1, and where condition (1.15) provides us with a loop type subcontinuum bifurcating both subcritically and supercritically at (0,0).
Theorem 1.2.
Under Bu=∂u∂n, assume (1.7), (1.8), (1.9), (1.10), (1.11), (1.16), (Hψ) with ψ=±a, and (Hb). Then, the following two assertions hold:
If (
1.15
) holds, then (
(PB)
) admits a loop type subcontinuum
𝒞
0
in
ℝ
×
C
1
(
Ω
¯
)
of nonnegative solutions for which the same assertions in Theorem
1.1
(i) hold true, see Figure 1 (i).
Assume (
1.1
), (
1.17
) and the condition that
Ω
+
a
is connected. Let
𝒞
∗
be the loop type subcontinuum stated above. If
q
∈
𝒜
ℬ
a
, then the same conclusion in Theorem
1.1
(ii) holds with 𝒞0 replaced by 𝒞∗.
Remark 1.3.
When ℬu=∂u∂𝐧 and ∫Ωb(x)≥0, the existence of a loop type subcontinuum of nonnegative solutions of (PB) has been established in the particular case f(s)=sq and g(s)=sp with 0<q<1<p (see [25], and as a particular case, see also (1.17)). In this case, although the loop type subcontinuum 𝒞∗ satisfies (1.6), 𝒞∗∖{(0,0)} appears in λ>0. This means that 𝒞∗ never meets the vertical line {(0,u):0≢u≥0}, see [23, Lemma 6.8 (1)]. Thus, the approach used in the proof of Theorem 1.2 (i) does not work for excluding the possibility that 𝒞∗={(0,0)}, see the argument in Subsection 5.1. Let us mention that in [25] the authors used a suitable rescaling technique (which strongly relies on the homogeneity of f(s)=sq and g(s)=sp) to exclude this possibility.
Remark 1.4.
Theorems 1.1 and 1.2 can be extended to the case a,b∈L∞(Ω) except assertion (ii) in Theorem 1.2. This can be done if we formulate (Hψ) for ψ∈L∞(Ω) such that ψ≢0, letting Ω+ψ be the largest open subset of Ω in which ψ>0 a.e., and assuming additionally
{
ψ
is bounded away from zero on compact subsets of
Ω
i
for
i
=
1
,
…
,
l
,
and
|
(
supp
ψ
+
)
∖
Ω
+
ψ
|
=
0
.
2 Preliminaries and Examples
We start this section with some remarks concerning some of our assumptions.
Remark 2.1.
Condition (1.8) implies:
By the L’Hospital rule,
(2.1)
lim
s
→
0
+
f
(
s
)
s
q
=
f
0
q
>
0
.
In particular, since f∈C1((0,∞)) and f(0)=0, we can show that f∈Cα([0,s0]) for α∈(0,q] and s0>0.
f satisfies the slope condition, that is, for any s0>0, there exists M0>0 such that
(2.2)
f
(
s
)
-
f
(
t
)
s
-
t
>
-
M
0
for
0
≤
t
<
s
≤
s
0
.
However, even under (1.8), a(x)f(⋅) does not satisfy the slope condition for x∈Ω where a(x)<0, since lims→0+f′(s)=∞.
Remark 2.2.
Since f(s)>0 for s>0, we note that if (1.9) holds, then f is concave for s>0, i.e.,
(
f
(
s
)
s
)
′
<
0
for
s
>
0
.
It is easy to check that (1.9) is stronger than (1.4). This is a consequence of the fact that (1.9) yields
0
≤
lim
s
→
∞
f
(
s
)
s
q
<
∞
.
Let us also note that (1.3) and (1.10) imply that g(s)>0 for s>0. Indeed, assume first g(s0)<0 for some s0>0, and set ε0:=-g(s0)/s0>0. From (1.3), we infer that for some s1∈(0,s0),
g
(
s
0
)
s
0
=
-
ε
0
<
g
(
s
1
)
s
1
,
which contradicts (1.10). Hence g(s)≥0 for all s>0. Next, assume that g(s0)=0 for some s0>0. By (1.10), it follows that g(s)≢0 for s∈(0,s0). This implies that g(s1)>0 for some s1∈(0,s0). It follows that
g
(
s
1
)
s
1
>
0
=
g
(
s
0
)
s
0
,
which contradicts (1.10) again, as desired.
Remark 2.3.
We describe here the explicit formula for the growth condition (Hb) of b+ in a neighborhood of ∂Ω+b used in Theorems 1.1 and 1.2, which originates from Amann and López-Gómez [2, Theorem 4.3]:
Ω
+
b
is a subdomain of Ω with smooth boundary ∂Ω+b and either
Ω
+
b
¯
⊂
Ω
and b<0 in Db:=Ω∖Ω+b¯, or
Ω
+
b
⊃
{
x
∈
Ω
:
d
(
x
,
∂
Ω
)
<
σ
}
for some σ>0 and b<0 in Db.
In addition, Db is a subdomain of Ω with smooth boundary, and there exist γ>0 and a function β defined in a tubular neighborhood U:={x∈Ω+b:d(x,∂Ω+b)<σ} of ∂Ω+b in Ω+b, which is continuous, positive and bounded away from zero, and satisfies b+(⋅)=β(⋅)d(⋅,∂Ω+b)γ in U and
1
<
p
<
min
(
N
+
2
N
-
2
,
N
+
1
+
γ
N
-
1
)
if
N
>
2
.
We conclude this section showing some examples of functions satisfying the previous conditions. We start with the following lemma, which characterizes the functions satisfying (1.8) and (1.9).
Lemma 2.4.
Let f∈C([0,∞))∩C1((0,∞)) with f(0)=0 and f(s)>0 for s>0. Then, the following two conditions are equivalent:
(
1.8
) and (
1.9
) hold.
f
(
s
)
=
s
q
h
(
s
)
for some
q
∈
(
0
,
1
)
and
h
∈
C
(
[
0
,
∞
)
)
∩
C
1
(
(
0
,
∞
)
)
such that
h
is nonincreasing
,
h
(
s
)
>
0
𝑓𝑜𝑟
s
≥
0
𝑎𝑛𝑑
lim
s
→
0
+
s
h
′
(
s
)
=
0
.
Proof.
It is easy to see that f as in condition (ii) fulfills (1.8) and (1.9). Conversely, if f satisfies the aforementioned conditions, defining h(s):=s-qf(s) for s>0 and h(0):=lims→0+h(s), it is also easy to check that h has the desired properties. ∎
As particular cases, we mention h(s)=11+sr (r≥0) and h(s)=e-s.
We note that oscillatory cases are out of our scope. For instance, consider h(s)=sin(1s)+2. If we put f(s):=sqh(s) for s>0 and f(0):=0, then f∈C([0,∞))∩C1((0,∞)) with f(0)=0 and f(s)>0 for s>0. Moreover, f fulfills (1.2) and (1.4), but (2.1), (2.2) and (1.9) do not hold.
We now exhibit examples of g satisfying (1.10), (1.11), and (1.16).
(a) We set
g
(
s
)
:
=
s
p
h
(
s
)
,
with
1
<
p
<
N
+
2
N
-
2
if
N
>
2
,
where h∈C1([0,∞)) is nondecreasing, bounded, and satisfies one of the following conditions:
0
=
h
(
0
)
<
h
′
(
0
)
.
Note that (1.16) holds if we choose σ:=p+1 in (i) and σ:=p in (ii). An example for (i) is h(s)=1-e-s, while h(s)=arctan(s+1) is included in (ii). Another example is given by h(s)=sr1+sr with r=0 or r=1. The case r=1 satisfies (i), whereas r=0 satisfies (ii). More generally, the function g(s)=sp+r1+sr (0≤r<2NN-2-p) fulfills (1.10), (1.11) and (1.16). Indeed, we can take σ:=p+r for (1.16).
(b) Let k≥1 and 1<p<N+2N-2 if N>2. We put
g
(
s
)
:
=
s
p
k
+
s
1
+
s
.
Then, g satisfies (1.11) and (1.16). It is also clear that (1.10) holds when k=1. Meanwhile, when k>1, it satisfies (1.10) if additionally
p
>
p
1
(
k
)
,
where
p
1
(
k
)
:
=
2
k
k
+
1
.
We note that p1(k) is increasing for k>1, and p1(k)↘1 as k→1+, whereas p1(k)↗2 as k→∞. Let us finally observe that case (b) is not included in any of the possibilities considered in case (a). Indeed, h(s)=k+s1+s is decreasing for s≥0.
3 Regularization Schemes and Transversality Conditions
Let us now explain our approach to study bifurcation of nontrivial nonnegative solutions for (PB) from (λ,0). From (1.2), we see that f is not differentiable at s=0, so that we can not directly apply the usual bifurcation theory from simple eigenvalues to (PB). To overcome this difficulty, we proceed as in [23, 25], “regularizing” (PB) at u=0, using ε>0. We refer to [19, Section 5] for a similar approach introducing a new parameter for a different regular problem.
We extend g to ℝ as a C1 function and set F:ℝ→ℝ by
F
(
s
)
:
=
{
s
1
-
q
f
(
s
)
,
s
≥
0
,
f
0
q
s
,
s
<
0
.
For ε>0, we shall study the auxiliary problem
(PcalBε)
{
-
Δ
u
=
λ
a
(
x
)
(
u
+
ε
)
q
-
1
F
(
u
)
+
b
(
x
)
g
(
u
)
in
Ω
,
ℬ
u
=
0
on
∂
Ω
.
Note that (1.8) and (2.1) imply that F∈C1(ℝ), F(0)=0 and F′(0)=f0q, so that
(3.1)
s
↦
(
s
+
ε
)
q
-
1
F
(
s
)
∈
C
1
(
(
-
ε
,
∞
)
)
.
Observe also that (Pℬ,0) corresponds to (PB), as far as nonnegative solutions are concerned.
Let us set
h
λ
,
ε
(
x
,
s
)
:
=
λ
a
(
x
)
(
s
+
ε
)
q
-
1
F
(
s
)
+
b
(
x
)
g
(
s
)
.
From (3.1), we see that h(x,⋅) satisfies the slope condition. Consequently, given a nontrivial nonnegative solution u of (PcalBε), we can choose M>0 such that (-Δ+M)u≥0 and ≢0 in Ω. Thus, by the strong maximum principle and Hopf’s lemma, we deduce u≫0, see [13], [18, Theorem 7.10].
We shall then consider the linearized eigenvalue problem at u=0 for the regular problem (PcalBε):
(3.2)
{
-
Δ
ϕ
=
λ
a
(
x
)
f
0
q
ε
q
-
1
ϕ
in
Ω
,
ℬ
ϕ
=
0
on
∂
Ω
.
Since a changes sign, (3.2) has exactly two principal eigenvalues λ1,ε-<0<λ1,ε+ (resp. λ1,ε-=0<λ1,ε+) if
ℬ
u
=
u
(
respectively
ℬ
u
=
∂
u
∂
𝐧
and
∫
Ω
a
<
0
)
,
which are both simple, and furthermore, (λ1,ε±,0) satisfy the Crandall–Rabinowitz transversality condition, see [18, Theorem 9.4].
Thanks to the simplicity and transversality condition, the local bifurcation theory [9, Theorem 1.7] ensures the existence and uniqueness of positive solutions of (PcalBε) bifurcating at (λ1,ε±,0). Moreover, the unilateral global bifurcation theory [17, Theorem 6.4.3] (see also [22, Theorem 1.27]) ensures that (PcalBε) possesses two components 𝒞ε±={(λ,u)} (i.e., maximal, nonempty, closed and connected subsets in ℝ×V) of nonnegative solutions emanating from (λ1,ε±,0), respectively (see Remark 3.2). In addition, 𝒞ε+∖{(λ1,ε±,0)} and 𝒞ε-∖{(λ1,ε±,0)} consist of positive solutions. This is due to elliptic regularity and the fact (see [1, Proposition 18.1]) that (PcalBε) has no bifurcating positive solutions from (λ,0) at any λ≠λ1,ε±.
Under some additional growth condition on g, we shall verify that 𝒞ε± are bounded in ℝ×V uniformly in ε∈(0,1], so that 𝒞ε-=𝒞ε+ (:=𝒞ε) (i.e. 𝒞ε is a mushroom). By simple computations, it can be shown easily that
(3.3)
λ
1
,
ε
±
→
0
as
ε
→
0
+
,
so that, passing to the limit as ε→0+, we shall observe by Whyburn’s topological argument [27, (9.12) Theorem] that
𝒞
0
:
=
lim sup
ε
→
0
+
𝒞
ε
is a loop type subcontinuum which consists of nonnegative solutions of (PB) and satisfies (1.6).
Remark 3.1.
The critical case
(3.4)
ℬ
u
=
∂
u
∂
𝐧
and
∫
Ω
a
=
0
can be handled in a similar way. In this case, we replace a by a-ε for ε>0 small in (PcalBε). Then, the above argument remains valid, since we can determine the asymptotic behavior (3.3) (see [23, Lemma 6.6] for the proof). In addition, we can reduce the case ∫Ωa>0 to the case ∫Ωa<0 under ℬu=∂u∂𝐧. Indeed, we only have to notice the symmetry property λa(x)=(-λ)(-a(x)). The situation may be illustrated by Figure 2 (i) and (ii).
Figure 2
The bounded component 𝒞ε for (PcalBε) with ℬu=∂u∂𝐧.
Remark 3.2.
We shall show that the transversality condition allows us to apply the unilateral global bifurcation result [17, Theorem 6.4.3] to (PcalBε) at (λ1,ε±,0). To this end, we reduce (PcalBε) to an operator equation in C(Ω¯). Given ξ∈Lr(Ω), r>N, let u∈Wℬ2,r(Ω) be the unique solution of
(3.5)
{
(
-
Δ
+
1
)
u
=
ξ
(
x
)
in
Ω
,
ℬ
u
=
0
on
∂
Ω
,
where
W
ℬ
2
,
r
(
Ω
)
:
=
{
{
u
∈
W
2
,
r
(
Ω
)
:
u
=
0
on
∂
Ω
}
if
ℬ
u
=
u
,
W
2
,
r
(
Ω
)
if
ℬ
u
=
∂
u
∂
𝐧
.
We introduce the solution operator 𝒮:Lr(Ω)→Wℬ2,r(Ω) associated with (3.5), implying that 𝒮(ξ)=u, which is bijective and homeomorphic. It follows that 𝒮:L∞(Ω)→Wℬ2,r(Ω) is continuous, and moreover, 𝒮:L∞(Ω)→V is compact, since so is the embedding Wℬ2,r(Ω)⊂V. Thus, (PcalBε) is reduced to
(3.6)
ℱ
(
λ
,
u
)
:
=
u
-
𝒮
[
(
λ
a
f
0
q
ε
q
-
1
+
1
)
u
+
h
λ
(
x
,
u
)
]
=
0
in
C
(
Ω
¯
)
,
where
h
λ
(
x
,
s
)
:
=
λ
a
(
x
)
{
(
s
+
ε
)
q
-
1
F
(
s
)
-
f
0
q
ε
q
-
1
s
}
+
b
(
x
)
g
(
s
)
.
Given u∈C(Ω¯), let
𝒜
λ
u
:
=
𝒮
[
(
λ
a
f
0
q
ε
q
-
1
+
1
)
u
]
,
and let
Ind
(
0
,
𝒜
λ
)
:
=
deg
(
1
-
𝒜
λ
,
B
R
)
be the fixed point index of 𝒜λ at the origin for λ≠λ1,ε± but close to λ1,ε±, where BR is the ball with radius R>0 and centered at the origin. According to [17, Section 4.2, Theorem 5.6.2], the transversality condition at (λ1,ε±,0) implies that Ind(0,𝒜λ) changes sign as λ crosses λ1,ε±. Thus, [17, Theorem 6.4.3] applies, so that equation (3.6) (and so, (PcalBε)) possesses two components 𝒞ε±={(λ,u)} in ℝ×C(Ω¯) of nonnegative solutions u emanating from (λ1,ε±,0), respectively. Finally, we can verify that 𝒞ε± are also components in ℝ×V, by elliptic regularity.
In the forthcoming sections, we will characterize the limiting behavior of 𝒞ε as ε→0+ under the conditions on a,b,f and g stated in Theorems 1.1 and 1.2.
4 A Priori Bounds
In this section, we establish an a priori bound for positive solutions of the regularized problem (PcalBε) in ℝ×V (Corollary 4.6).
We start with an a priori bound on λ∈ℝ, uniformly in ε∈[0,1].
Proposition 4.1.
Assume (1.3), (1.5), (1.8) and (1.7). Then, there exist λ¯,ε¯>0 such that if 0≤ε≤ε¯, then (PcalBε) has no positive solution for |λ|≥λ¯.
Proof.
Let us suppose first that we are not in the case (3.4). Let 0≤ε≤1, and assume that (PcalBε) has a positive solution u for some λ>0. Let B be given by (1.7), λB>0 be the first eigenvalue of the problem
(4.1)
{
-
Δ
ϕ
=
λ
ϕ
in
B
,
ϕ
=
0
on
∂
B
,
and ϕB∈C2(B¯) be a positive eigenfunction associated to λB. We extend ϕB to Ω¯ by setting ϕB=0 in Ω¯∖B¯, so that ϕB∈H01(Ω). Since u>0 on B¯ and ∂ϕB∂ν<0 on ∂B, the divergence theorem yields that
λ
B
∫
B
ϕ
B
u
=
∫
B
-
Δ
ϕ
B
u
=
∫
B
∇
ϕ
B
∇
u
-
∫
∂
B
∂
ϕ
B
∂
ν
u
>
∫
B
∇
ϕ
B
∇
u
,
where ν is the outward unit normal to ∂B. On the other hand, we see that
∫
B
∇
u
∇
ϕ
B
=
∫
B
b
(
x
)
g
(
u
)
ϕ
B
+
λ
∫
B
a
(
x
)
(
u
+
ε
)
q
-
1
F
(
u
)
ϕ
B
,
where q is given by (1.8). It follows that
(4.2)
∫
B
u
q
ϕ
B
{
b
(
x
)
g
(
u
)
u
q
+
λ
a
(
x
)
(
u
u
+
ε
)
1
-
q
F
(
u
)
u
-
λ
B
u
1
-
q
}
<
0
.
Now, for (x,s)∈B×(0,∞), we set
h
(
x
,
s
)
:
=
b
(
x
)
g
(
s
)
s
q
+
λ
a
(
x
)
(
s
s
+
ε
)
1
-
q
F
(
s
)
s
-
λ
B
s
1
-
q
.
By (1.5), there exists s0>0 such that
g
(
s
)
≥
λ
B
b
0
s
for
s
>
s
0
,
where b0 is from (1.7). Hence, since f(s)>0 for s>0 and a(x)≥0 a.e. in B, we deduce that if λ>0, x∈B and s>s0, then
(4.3)
h
(
x
,
s
)
≥
b
(
x
)
g
(
s
)
s
q
-
λ
B
s
1
-
q
≥
s
1
-
q
(
b
0
g
(
s
)
s
-
λ
B
)
≥
0
.
Let us now consider the case 0<s≤s0. From (1.3), we can choose K0>0 such that
|
g
(
s
)
s
|
≤
K
0
for
0
<
s
≤
s
0
.
Recalling (1.8) (or (2.1)), we set
M
1
:
=
inf
0
<
s
≤
s
0
F
(
s
)
s
>
0
.
By putting b∞:=∥b∥∞, it follows that
h
(
x
,
s
)
=
s
1
-
q
{
b
(
x
)
g
(
s
)
s
+
λ
a
(
x
)
(
1
s
+
ε
)
1
-
q
F
(
s
)
s
-
λ
B
}
≥
s
1
-
q
{
λ
a
0
(
1
s
0
+
1
)
1
-
q
M
1
-
(
λ
B
+
b
∞
K
0
)
}
,
so that h(x,s)≥0 for x∈B and 0<s≤s0 if
(4.4)
λ
≥
λ
¯
:
=
(
λ
B
+
b
∞
K
0
)
(
s
0
+
1
)
1
-
q
a
0
M
1
.
Consequently, by (4.2), (4.3) and (4.4) we deduce that λ<λ¯.
Next, let us verify the existence of a lower bound on λ<0 for the existence of a positive solution of (PcalBε). In order to check this, we notice that if (PcalBε) has a positive solution u for some λ<0 and ε∈[0,1], then
-
Δ
u
=
(
-
λ
)
(
-
a
(
x
)
)
(
u
+
ε
)
q
-
1
F
(
u
)
+
b
(
x
)
g
(
u
)
in
Ω
.
From (1.7), the desired conclusion follows arguing as above with B now replaced by B′.
It remains to consider case (3.4). However, it suffices to note that (1.7) implies that if ε is small enough, then
a
ε
(
x
)
≥
a
0
2
in
B
and
-
a
ε
(
x
)
≥
a
0
′
in
B
′
.
The proof now follows in the same way as above. ∎
Next, given a compact interval I, we establish an a priori upper bound for positive solutions of (PcalBε) whenever λ∈I and ε∈[0,1]. We start with the following preliminary lemma (see also [2, Theorem 4.1]).
Lemma 4.2.
Assume (1.5), (1.8), (1.9), (1.10) and (Hb). Let Λ>0. Suppose there exists a constant C1>0 such that ∥u∥C(Ω+b¯)≤C1 for all positive solutions u of (PcalBε) with λ∈[-Λ,Λ] and ε∈[0,1]. Then, there exists C2>0 such that ∥u∥C(Ω¯)≤C2 for all positive solutions u of (PcalBε) with λ∈[-Λ,Λ] and ε∈[0,1].
Proof.
(i) First, we consider the Dirichlet case. We use a comparison principle for concave problems inspired by the one in [3, Lemma 3.3].
(1) Assume first that Ω+b¯⊂Ω, and recall that Db is given by (Hb). Let λ∈[0,Λ], and consider the problem
(4.5)
{
-
Δ
v
=
λ
a
+
(
x
)
(
v
+
ε
)
q
-
1
F
(
v
)
-
b
-
(
x
)
g
(
v
)
in
D
b
,
v
=
C
1
on
∂
Ω
+
b
,
v
=
0
on
∂
Ω
.
Let u be a positive solution of (PcalBε), with λ∈[0,Λ] and ε∈[0,1]. It follows that u>0 in Db. Since λ≥0, b=-b- in Db, and f(s)>0 for s>0, it is easy to check that u is a subsolution of (4.5).
Next, we construct a supersolution of (4.5). Consider the unique positive solution w0 of the problem
{
-
Δ
w
=
1
in
D
b
,
w
=
0
on
∂
Ω
+
b
∪
∂
Ω
.
Set w¯:=C(w0+1) for C>0. If C≥C1, then w¯≥C1 on ∂Ω+b and w¯≥0 on ∂Ω. Moreover, we claim that if C is sufficiently large, then
-
Δ
w
¯
≥
λ
a
+
(
x
)
(
w
¯
+
ε
)
q
-
1
F
(
w
¯
)
-
b
-
(
x
)
g
(
w
¯
)
in
D
b
.
Indeed, let
δ
:
=
1
Λ
∥
a
+
∥
∞
(
∥
w
0
∥
C
(
D
b
¯
)
+
1
)
>
0
.
Then, from (1.9) and (1.5), there exists s1>0 large enough such that if s≥s1, then
0
≤
g
(
s
)
and
f
(
s
)
≤
δ
s
(
so that
F
(
s
)
≤
δ
s
2
-
q
)
.
It follows that if C≥s1, then
-
Δ
w
¯
-
{
λ
a
+
(
x
)
(
w
¯
+
ε
)
q
-
1
F
(
w
¯
)
-
b
-
(
x
)
g
(
w
¯
)
}
≥
C
-
Λ
∥
a
+
∥
∞
w
¯
q
-
1
δ
w
¯
2
-
q
≥
C
{
1
-
δ
Λ
∥
a
+
∥
∞
(
∥
w
0
∥
C
(
D
b
¯
)
+
1
)
}
=
0
in
D
b
.
Thus the claim has been verified, and w¯ is a supersolution of (4.5) if C≥max(C1,s1). Note that C can be chosen independently of λ∈[0,Λ] and ε∈[0,1].
Now, we see from (1.9) and (1.10) that the nonlinearity in (4.5) is concave, that is, if we set
j
(
x
,
s
)
:
=
λ
a
+
(
x
)
(
s
+
ε
)
q
-
1
F
(
s
)
-
b
-
(
x
)
g
(
s
)
,
x
∈
D
b
,
s
>
0
,
then
∂
∂
s
(
j
(
x
,
s
)
s
)
<
0
for
x
∈
D
b
,
s
>
0
.
Indeed,
d
d
s
(
(
s
+
ε
)
q
-
1
F
(
s
)
s
)
=
(
q
-
1
)
(
s
+
ε
)
q
-
2
(
s
-
q
f
(
s
)
)
+
(
s
+
ε
)
q
-
1
d
d
s
(
s
-
q
f
(
s
)
)
<
0
.
Reasoning as in [24, Proposition A.1] (whose argument is based on [3, Lemma 3.3]), we may deduce that u≤w¯ in Db, so that
u
≤
C
1
+
∥
w
¯
∥
C
(
D
b
¯
)
on
Ω
¯
.
It remains to verify the case -Λ≤λ<0. Note that any positive solution u of (PcalBε) with λ∈[-Λ,0) and ε∈[0,1] satisfies
-
Δ
u
=
μ
(
-
a
(
x
)
)
(
u
+
ε
)
q
-
1
F
(
u
)
+
b
(
x
)
g
(
u
)
in
D
b
,
with μ:=-λ∈(0,Λ]. Instead of (4.5), we consider the following concave problem:
(4.6)
{
-
Δ
v
=
μ
a
-
(
x
)
(
v
+
ε
)
q
-
1
F
(
v
)
-
b
-
(
x
)
g
(
v
)
in
D
b
,
v
=
C
1
on
∂
Ω
+
b
,
v
=
0
on
∂
Ω
.
Then, we see that u is a subsolution of this problem. The remainder of the argument is identical to the one in the case λ∈[0,Λ].
(2) Assume now that Ω+b⊃{x∈Ω:d(x,∂Ω)<σ} for some σ>0. This case can be verified identically. Indeed, it suffices to replace (4.5) by the problem
{
-
Δ
v
=
λ
a
+
(
x
)
(
v
+
ε
)
q
-
1
F
(
v
)
-
b
-
(
x
)
g
(
v
)
in
D
b
,
v
=
C
1
on
∂
Ω
+
b
∩
Ω
.
(ii) Lastly, we verify the Neumann case. Since aε≤a+ and -aε≤a-+1, it suffices to replace (4.5) by
{
-
Δ
v
=
λ
a
+
(
x
)
(
v
+
ε
)
q
-
1
F
(
v
)
-
b
-
(
x
)
g
(
v
)
in
D
b
,
v
=
C
1
on
∂
Ω
+
b
,
∂
v
∂
𝐧
=
0
on
∂
Ω
,
and (4.6) by
{
-
Δ
v
=
μ
(
a
-
(
x
)
+
1
)
(
v
+
ε
)
q
-
1
F
(
v
)
-
b
-
(
x
)
g
(
v
)
in
D
b
,
v
=
C
1
on
∂
Ω
+
b
,
∂
v
∂
𝐧
=
0
on
∂
Ω
.
The proof of Lemma 4.2 is now complete. ∎
The following result is due to Gidas and Spruck [12, Theorem 1.1] and Amann and López-Gómez [2, Section 4] (see also López-Gómez, Molina-Meyer and Tellini [20, Section 6]).
Proposition 4.3.
Assume (1.4), (1.8) and (1.11). In addition, suppose either
b
>
0
on
Ω
¯
and
p
<
N
+
2
N
-
2
for the Dirichlet case (
p
<
N
+
1
N
-
1
for the Neumann case) if
N
>
2
, or
(
1.9
), (
1.10
) and
(Hb)
hold.
Then, given Λ>0, there exists C>0 such that if 0≤ε≤1, then u≤C in Ω for all positive solutions u of (PcalBε) with |λ|≤Λ.
Remark 4.4.
Condition (i) for the Neumann case is based on (Hb) with γ=0 from Amann and López-Gómez [2, Section 4], not on Gidas and Spruck [12, Theorem 1.1].
Under condition (ii), we can handle the case where b>0 in Ω and b=0 somewhere on ∂Ω.
Proof.
Case (i) under ℬu=u is verified in the same way as in the proof of [12, Theorem 1.1].
Case (i) under ℬu=∂u∂𝐧 and case (ii) can be handled using the arguments in [23, Proposition 6.5] and [24, Proposition 4.2], which are based on [20, Section 6]. Indeed, based on (1.4), (1.11) and (Hb), we employ the argument developed by Amann and López-Gómez [2] to deduce that the hypothesis of Lemma 4.2 is fulfilled, so that the desired conclusion follows. ∎
By ∥⋅∥H we denote the usual norm of H=H01(Ω) when ℬu=u and of H=H1(Ω) when ℬu=∂u∂𝐧. The next lemma follows easily by a bootstrap argument based on elliptic regularity and the Sobolev embedding theorem.
Lemma 4.5.
Assume that there exist C>0 and p∈(1,N+2N-2) such that
|
g
(
s
)
|
≤
C
(
1
+
s
p
)
𝑓𝑜𝑟
s
≥
0
.
Let Λ>0 and u be a nonnegative solution of (PcalBε) for λ∈[-Λ,Λ] and ε∈[0,1]. Then, given c1>0, there exists c2>0 such that if ∥u∥H≤c1, then ∥u∥V≤c2.
With the aid of Lemma 4.5, Propositions 4.1 and 4.3 provide us with an a priori bound in ℝ×V for positive solutions of (PcalBε) uniformly in ε∈[0,1].
Corollary 4.6.
Under the assumptions of Propositions 4.1 and 4.3, there exists ε¯,C¯>0 such that if 0≤ε≤ε¯, then |λ|+∥u∥V≤C¯ for all positive solutions u of (PcalBε).
Proof.
Propositions 4.1 and 4.3 imply that there exist ε¯,C>0 such that if 0≤ε≤ε¯, then |λ|+∥u∥H≤C for all positive solutions u of (PcalBε), since u satisfies
∫
Ω
|
∇
u
|
2
=
λ
∫
Ω
{
a
(
x
)
(
u
+
ε
)
q
-
1
F
(
u
)
u
+
b
(
x
)
g
(
u
)
u
}
≤
C
′
for some C′>0. Lemma 4.5 provides then the desired conclusion. ∎
We discuss now bifurcation of nontrivial nonnegative solutions for (PB) from (λ,0) for λ>0. We prove the following preliminary lemma.
Lemma 4.7.
Assume (1.8) and (Hψ) with ψ=a. Let Λ>0, Ω′ be a nonempty connected open subset of Ω+a, and B⋐Ω′ be a ball. Then there exists CΛ>0 such that if λ≥Λ, then ∥u∥C(B¯)≥CΛ for all nontrivial nonnegative solutions u of (PB) such that u≢0 in Ω′.
Proof.
We use an argument based on subsolutions and supersolutions. First of all, we remark that (1.8) implies (1.2) and (2.2), see Remark 2.1.
Let Λ,Ω′ and B be as in the statement of this lemma, and let λ≥Λ. Assume that u is a nontrivial nonnegative solution of (PB) such that u≢0 in Ω′. Set
(4.7)
K
1
:
=
max
0
≤
s
≤
∥
u
∥
C
(
Ω
¯
)
|
g
′
(
s
)
|
≥
0
.
Since g(0)=0, the mean value theorem provides some constant θ=θx∈(0,1) such that g(u)=g′(θu)u. Thus, using (4.7) we get that
(
-
Δ
+
b
∞
K
1
+
1
)
u
=
λ
a
(
x
)
f
(
u
)
+
(
b
∞
K
1
+
1
+
b
(
x
)
g
′
(
θ
u
)
)
u
≥
u
≥
0
and
≢
0
in
Ω
′
.
The strong maximum principle yields that u>0 in Ω′. Now, let s0>0 be fixed. Then, the following two possibilities may occur: (i) u≤s0 on B¯; (ii) u>s0 somewhere on B¯.
We consider case (i). We have a0:=infB¯a>0, so that u is a supersolution of the problem
(4.8)
{
-
Δ
v
+
b
∞
K
2
v
=
λ
a
0
f
(
v
)
in
B
,
v
=
0
on
∂
B
,
where
K
2
:
=
max
0
≤
s
≤
s
0
|
g
′
(
s
)
|
≥
0
.
Indeed, u≥0 on ∂B. Moreover, since f(s)>0 for s>0, the mean value theorem shows that
-
Δ
u
+
b
∞
K
2
u
-
λ
a
0
f
(
u
)
=
λ
(
a
(
x
)
-
a
0
)
f
(
u
)
+
(
b
∞
K
2
+
b
(
x
)
g
′
(
θ
u
)
)
u
≥
0
in
B
.
To construct a subsolution of (4.8), we use the positive eigenfunction ϕB associated to the first eigenvalue λB of (4.1) and such that ∥ϕB∥C(B¯)=1. From (1.2), we find a constant s1>0 small enough such that
(4.9)
f
(
s
)
s
≥
λ
B
+
b
∞
K
2
Λ
a
0
for
0
<
s
≤
s
1
.
If 0<s≤s1, then we observe that
-
Δ
(
s
ϕ
B
)
+
b
∞
K
2
s
ϕ
B
-
λ
a
0
f
(
s
ϕ
B
)
≤
s
ϕ
B
{
λ
B
+
b
∞
K
2
-
Λ
a
0
f
(
s
ϕ
B
)
s
ϕ
B
}
≤
0
in
B
.
This implies that sϕB is a subsolution of (4.8) whenever 0<s≤s1. Now, since u>0 in Ω′, it follows that u>0 on B¯. Furthermore, we assert that
(4.10)
u
≥
s
1
ϕ
B
on
B
¯
.
By contradiction, we assume that u≱s1ϕB. Then, since u>0=s1ϕB on ∂B, we can choose σ∈(0,1) such that u-σs1ϕB≥0 on B¯ and u-σs1ϕB=0 somewhere in B. From (2.2), we fix M0>0 such that
f
(
s
)
-
f
(
t
)
s
-
t
>
-
M
0
for
0
≤
t
<
s
≤
s
0
.
Putting M1:=λa0M0>0, we see that the mapping
s
↦
M
1
s
+
λ
a
0
f
(
s
)
is nondecreasing for 0≤s≤s0. Indeed, if 0≤t<s≤s0, then
M
1
s
+
λ
a
0
f
(
s
)
-
(
M
1
t
+
λ
a
0
f
(
t
)
)
=
(
M
1
+
λ
a
0
f
(
s
)
-
f
(
t
)
s
-
t
)
(
s
-
t
)
≥
(
M
1
-
λ
a
0
M
0
)
(
s
-
t
)
=
0
,
as desired. Thus, using this monotonicity and having in mind that (recall (4.9))
(
-
Δ
+
b
∞
K
2
)
σ
s
1
ϕ
B
≤
λ
a
0
f
(
σ
s
1
ϕ
B
)
in
B
,
we deduce that
(
-
Δ
+
b
∞
K
2
+
M
1
)
(
u
-
σ
s
1
ϕ
B
)
≥
M
1
u
+
λ
a
0
f
(
u
)
-
(
M
1
σ
s
1
ϕ
B
+
λ
a
0
f
(
σ
s
1
ϕ
B
)
)
≥
0
in
B
and u-σs1ϕB=u>0 on ∂B. The strong maximum principle yields that u-σs1ϕB>0 in B, a contradiction. Thus, we have verified (4.10). By taking into account case (ii), CΛ:=min{s0,s1} is as desired. ∎
By virtue of Lemma 4.7, there are no nontrivial nonnegative solutions of (PB) bifurcating from (λ,0) for λ>0, and moreover, there exist no small positive solutions of (PB) for λ=0. In view of this fact, although we shall observe that (PB) possesses a bounded subcontinuum of nontrivial nonnegative solutions bifurcating at (0,0), we infer that the bifurcation subcontinuum is of loop type.
Proposition 4.8.
Assume (1.3). Then, the following assertions hold:
Assume (
(Hψ)
) with
ψ
=
a
, and additionally (
1.8
) if
ℬ
u
=
∂
u
∂
𝐧
. Then, given
λ
0
>
0
, there exist
δ
0
,
c
0
>
0
such that
∥
u
∥
C
(
Ω
¯
)
≥
c
0
for all nontrivial nonnegative solutions
u
of (
(PB)
) for
λ
∈
(
λ
0
-
δ
0
,
λ
0
+
δ
0
)
.
Assume (
1.15
) and (
1.16
) if
ℬ
u
=
∂
u
∂
𝐧
. Then, there exists
C
>
0
such that
∥
u
∥
C
(
Ω
¯
)
≥
C
for all positive solutions
u
of (
(PB)
) for
λ
=
0
.
Proof.
(i) We recall that (1.8) also implies (2.1), see Remark 2.1. First, we verify the Dirichlet case. By contradiction, we assume that λn→λ0>0, and un are nontrivial nonnegative solutions of (PB) with λ=λn such that ∥un∥C(Ω¯)→0. Then, we claim that, up to a subsequence, ∫Ωa(x)f(un)un≤0. If not, then we may suppose that ∫Ωa(x)f(un)un>0 for all n. It follows that un≢0 in Ω+a. Indeed, if un≡0 in Ω+a, then, using that f(s)>0 for s>0, we find that
0
<
∫
Ω
a
(
x
)
f
(
u
n
)
u
n
≤
∫
Ω
+
a
a
(
x
)
f
(
u
n
)
u
n
=
0
,
which is a contradiction. Employing (Hψ) with ψ=a, we may deduce that there exists a connected open subset Ω′⊂Ω+a such that un≢0 in Ω′ for all n≥1. Let B⋐Ω′ be a ball. We apply Lemma 4.7 with Λ=λ02, to derive that ∥un∥C(B¯)≥c0 for some c0>0 independent of n, which contradicts ∥un∥C(Ω¯)→0. Thus, the claim follows.
Now, we observe from the definition of un that
∥
u
n
∥
H
2
:
=
∫
Ω
|
∇
u
n
|
2
=
λ
n
∫
Ω
a
(
x
)
f
(
u
n
)
u
n
+
∫
Ω
b
(
x
)
g
(
u
n
)
u
n
≤
∫
Ω
b
∞
|
g
(
u
n
)
|
u
n
.
Set vn:=un/∥un∥H, so that
(4.11)
∥
v
n
∥
H
2
≤
b
∞
∫
Ω
|
g
(
u
n
)
|
u
n
∥
u
n
∥
H
2
.
From (1.3), given ε>0, there exists sε>0 such that
|
g
(
s
)
|
≤
ε
b
∞
s
for
0
<
s
≤
s
ε
.
Also, for n large enough, we have that ∥un∥C(Ω¯)≤sε, so that
(4.12)
b
∞
∫
Ω
|
g
(
u
n
)
|
u
n
∥
u
n
∥
H
2
≤
ε
∫
Ω
v
n
2
.
From (4.11) and (4.12), we derive that vn→0 in H01(Ω), a contradiction.
Next, we verify the Neumann case. Assume to the contrary that λn→λ0>0, and un are nontrivial nonnegative solutions of (PB) with λ=λn such that ∥un∥C(Ω¯)→0. We remark that ∥un∥H→0, since un are nonnegative solutions of (PB) with λ=λn. Arguing as in the proof for the Dirichlet case, we have that, up to a subsequence, ∫Ωa(x)f(un)un≤0, and consequently, ∫Ω|∇un|2≤b∞∫Ω|g(un)|un.
Set vn:=un/∥un∥H, so that ∥vn∥H=1. We may assume that there exists v0∈H1(Ω) such that vn⇀v0 in H1(Ω), vn→v0 a.e. in Ω, and vn→v0 in Lt(Ω) for t<2*. By (1.3), for any ε>0, there exists s0>0 such that
(4.13)
|
g
(
s
)
|
≤
ε
b
∞
s
for
0
≤
s
≤
s
0
.
Thus, for n large enough, we have that ∥un∥C(Ω¯)≤s0, so that
∫
Ω
|
∇
v
n
|
2
≤
b
∞
∫
Ω
|
g
(
u
n
)
|
∥
u
n
∥
H
v
n
≤
ε
∫
Ω
v
n
2
≤
ε
.
This implies that ∫Ω|∇vn|2→0, and it follows that vn→v0, and v0 is a positive constant.
Since un is a nonnegative solution of (PB) with λ=λn, we see that, for every ϕ∈H1(Ω),
(4.14)
(
∫
Ω
∇
v
n
∇
ϕ
)
∥
u
n
∥
H
1
-
q
=
λ
n
∫
Ω
a
(
x
)
f
(
u
n
)
∥
u
n
∥
H
q
ϕ
+
∫
Ω
b
(
x
)
g
(
u
n
)
∥
u
n
∥
H
q
ϕ
.
Since ∥un∥C(Ω¯)→0, (4.13) implies that |g(un)|≤un for n large enough, so that
|
∫
Ω
b
(
x
)
g
(
u
n
)
∥
u
n
∥
H
q
ϕ
|
≤
b
∞
∫
Ω
u
n
∥
u
n
∥
H
q
|
ϕ
|
≤
b
∞
∥
u
n
∥
H
1
-
q
(
∫
Ω
v
n
2
)
1
2
(
∫
Ω
ϕ
2
)
1
2
→
0
.
We use this inequality to deduce from (4.14) that, passing to the limit as n→∞,
∫
Ω
a
(
x
)
f
(
u
n
)
∥
u
n
∥
H
q
ϕ
→
0
.
On the other hand, since f(0)=0, we have that
∫
Ω
a
(
x
)
f
(
u
n
)
∥
u
n
∥
H
q
ϕ
=
∫
v
n
>
0
a
(
x
)
f
(
∥
u
n
∥
H
v
n
)
(
∥
u
n
∥
H
v
n
)
q
v
n
q
ϕ
.
Thus, using (2.1) and the fact that un→0 in H1(Ω), vn→v0 in Lt(Ω), vn→v0 a.e. in Ω and v0 is a positive constant, the Lebesgue dominated convergence theorem yields that
∫
Ω
a
(
x
)
f
(
u
n
)
∥
u
n
∥
H
q
ϕ
→
f
0
q
v
0
q
∫
Ω
a
(
x
)
ϕ
.
Therefore,
∫
Ω
a
(
x
)
ϕ
=
0
.
Since ϕ∈H1(Ω) is arbitrary, we find that a≡0, which is a contradiction.
(ii) In the Dirichlet case, we argue as in the proof of assertion (i) to prove assertion (ii), by taking λn=λ0=0 therein.
Next, we verify the Neumann case. Assume to the contrary that there exist positive solutions un of (PB) for λ=0 such that ∥un∥C(Ω¯)→0. Then, as in the proof of assertion (i), we may deduce from (1.3) that vn:=un/∥un∥H1(Ω)→v0 in Lt(Ω) for t<2*, and v0 is a positive constant. Since un is a positive solution of (PB) with λ=0, we obtain ∫Ωb(x)g(un)=0. Recalling (1.16), we see that
0
=
∫
Ω
b
(
x
)
g
(
u
n
)
∥
u
n
∥
H
1
(
Ω
)
σ
→
g
0
v
0
σ
∫
Ω
b
(
x
)
,
so that ∫Ωb(x)=0, which contradicts (1.15).
The proof is now complete. ∎
Assuming additionally (Hψ) with ψ=-a, we can extend Proposition 4.8 (i) to λ<0, and in this case, bifurcation of nontrivial nonnegative solutions of (PB) from (λ,0) can only occur at (0,0).
Corollary 4.9.
Under the assumptions of Proposition 4.8, assume in addition (Hψ) with ψ=-a. Then the conclusion of Proposition 4.8 (i) holds for all λ0≠0. In particular, given δ∈(0,1), the set of nontrivial nonnegative solutions of (PB) is away from the set {(λ,0);δ≤|λ|≤δ-1}.
Proof.
In view of Proposition 4.8, it remains to verify the case λ<0. Assume to the contrary that λn→λ0<0, and un are nontrivial nonnegative solutions of (PB) with λ=λn such that un→0 in C(Ω¯). Then, we have that
-
Δ
u
n
=
(
-
λ
n
)
(
-
a
(
x
)
)
f
(
u
n
)
+
b
(
x
)
g
(
u
n
)
in
Ω
.
By the same arguments used in Lemma 4.7 and Proposition 4.8, we get the desired conclusion. ∎
5 Proofs of Theorems 1.1 and 1.2
If X is a metric space and En⊂X, then we set
lim inf
n
→
∞
E
n
:
=
{
x
∈
X
:
lim
n
→
∞
dist
(
x
,
E
n
)
=
0
}
,
lim sup
n
→
∞
E
n
:
=
{
x
∈
X
:
lim inf
n
→
∞
dist
(
x
,
E
n
)
=
0
}
.
We shall use the following result due to Whyburn [27, (9.12) Theorem]:
Theorem 5.1.
Assume {En} is a sequence of connected sets satisfying that
⋃
n
≥
1
E
n
is precompact;
lim inf
n
→
∞
E
n
≠
∅
.
Then, lim supn→∞En is nonempty, closed and connected.
As stated in Remark 3.1, we only have to prove Theorem 1.2 (i) in the case ∫Ωa<0.
5.1 Proof of Assertion (i) in Theorems 1.1 and 1.2
When ℬu=∂u∂𝐧, we employ the following lemma, which concerns the direction of the bifurcation component 𝒞ε at (0,0), see [23, Theorem 5.1] for the proof.
Lemma 5.2.
Let Bu=∂u∂n. Assume (1.8), (1.16) and ∫Ωa≠0. Let Z be any complement of 〈1〉 in W2,r(Ω). Then, for ε>0 small enough, the set {(λ,u)} of nontrivial solutions of (PcalBε) around (0,0) is parameterized as
(
λ
,
u
)
=
(
μ
(
s
)
,
s
(
1
+
z
(
s
)
)
)
,
with s∈(-s0,s0) for some s0>0. Here, μ:(-s0,s0)→R and z:(-s0,s0)→Z are continuous, and satisfy μ(0)=z(0)=0. Therefore, Cε is precisely described by {(μ(s),s(1+z(s))):s∈[0,s0)} around (0,0). Furthermore, the following holds:
(5.1)
lim
s
→
0
+
μ
(
s
)
s
σ
-
1
=
-
ε
1
-
q
q
g
0
∫
Ω
b
(
x
)
f
0
∫
Ω
a
(
x
)
.
In particular, under (1.15) and ∫Ωa<0, the bifurcation of Cε is subcritical at (0,0).
Now, we consider the metric space X:=ℝ×V with the metric given by
d
(
(
λ
,
u
)
,
(
μ
,
v
)
)
:
=
|
λ
-
μ
|
+
∥
u
-
v
∥
V
for
(
λ
,
u
)
,
(
μ
,
v
)
∈
ℝ
×
V
.
From Corollary 4.6, if ε∈(0,1], then the components 𝒞ε± of positive solutions of (PcalBε), emanating from (λ1,ε±,0), satisfy
(5.2)
𝒞
ε
±
⊂
{
(
λ
,
u
)
∈
ℝ
×
V
:
|
λ
|
+
∥
u
∥
V
≤
C
¯
}
,
where C¯ does not depend on ε∈(0,1]. This implies that 𝒞ε± are both bounded, and consequently, we deduce that 𝒞ε-=𝒞ε+ (cf. [1, Proposition 18.1]). Then, 𝒞ε:=𝒞ε± is nonempty and connected. In addition,
(5.3)
(
0
,
0
)
∈
lim inf
ε
→
0
+
𝒞
ε
,
since λ1,ε±→0 as ε→0+. Moreover, by elliptic regularity, we obtain that
(5.4)
⋃
ε
>
0
𝒞
ε
is precompact
.
Indeed, for any {(λn,un)}⊂⋃ε>0𝒞ε, we have that (λn,un)∈𝒞εn for some εn∈(0,1]. From (5.2), we may assume that {λn} is a convergent sequence. Using (5.2) again, we deduce that un∈W2,r(Ω) are solutions of
{
-
Δ
u
n
=
λ
n
a
(
x
)
(
u
n
+
ε
n
)
q
-
1
F
(
u
n
)
+
b
(
x
)
g
(
u
n
)
in
Ω
,
u
n
=
0
on
∂
Ω
.
In particular, using a bootstrap argument and the Sobolev embedding theorem, we deduce that ∥un∥C1+θ(Ω¯) is bounded for some θ∈(0,1). The compact embedding C1+θ(Ω¯)⊂C1(Ω¯) implies that {un} has a convergent subsequence in V, as desired. Now, by (5.3) and (5.4), we may apply Theorem 5.1 to infer that 𝒞0:=lim supε→0+𝒞ε is nonempty, closed and connected in ℝ×V. From (5.2), 𝒞0 is bounded in ℝ×V. In addition, 𝒞0 is contained in the nonnegative solutions set of (PB). Indeed, given (λ,u)∈𝒞0, there exists (λn,un)∈𝒞εn such that εn→0+ and (λn,un)→(λ,u) in ℝ×V. Thus u is a nonnegative weak solution of (PB), and eventually, a nonnegative solution in W2,r(Ω) by elliptic regularity.
Now, we show that 𝒞0 is nontrivial. By construction, we see that for ε→0+, there exists a positive solution uε of (PcalBε) such that (0,uε)∈𝒞ε. Indeed, we used Lemma 5.2 if ℬu=∂u∂𝐧. In this case, we observe from (5.1) that when (1.15) and ∫Ωa<0 hold, the bounded component 𝒞ε bifurcates subcritically at (0,0), provided that ε is small enough. This implies that 𝒞ε cuts {(0,u):0≢u≥0}, and consequently, the desired assertion follows. Since ∥uε∥V≤C¯, it follows by combining elliptic regularity and standard compactness arguments as above that there exist εn→0+ and un:=uεn such that un converges in V to a nonnegative solution u0 of (PB) for λ=0. By definition, we have that (0,u0)∈𝒞0. From Proposition 4.8 (ii), we infer that u0 is nontrivial, and so, u0≫0 by the strong maximum principle and Hopf’s lemma. Assertion (i) (1) has been now verified. We use Proposition 4.8 (ii) again to deduce assertion (i) (2).
Since 𝒞0 is nontrivial, we infer from Corollary 4.9 that 𝒞0 does not contain any (λ,0) with λ≠0. Assertion (1.6) has been verified.
Finally, we verify assertion (i) (3). For ρ>0 and (λ1,u1)∈ℝ×V, we set
B
ρ
(
(
λ
1
,
u
1
)
)
:
=
{
(
λ
,
u
)
∈
ℝ
×
V
:
|
λ
-
λ
1
|
+
∥
u
-
u
1
∥
V
<
ρ
}
,
S
ρ
(
(
λ
1
,
u
1
)
)
:
=
{
(
λ
,
u
)
∈
ℝ
×
V
:
|
λ
-
λ
1
|
+
∥
u
-
u
1
∥
V
=
ρ
}
.
We note that Bρ((λ1,u1))¯=Bρ((λ1,u1))∪Sρ((λ1,u1)).
Let Σε+ and Σε- be closed connected subsets of {(λ,u)∈𝒞ε:λ≥0} and {(λ,u)∈𝒞ε:λ≤0}, respectively, such that (λ1,ε±,0),(0,uε±)∈Σε± for some positive solutions uε± of (PB) for λ=0, see Figure 3. This is well defined thanks to Proposition 4.8 (ii). Since Σε±⊂𝒞ε, we observe that
Σ
0
±
:
=
lim sup
ε
→
0
+
Σ
ε
±
⊂
lim sup
ε
→
0
+
𝒞
ε
=
𝒞
0
.
Repeating the argument above, Whyburn’s topological approach yields that Σ0± are nonempty, closed and connected sets consisting of nonnegative solutions of (PB) and such that (0,0)∈lim infε→0+Σε±⊂Σ0±. Proposition 4.8 (ii) tells us that (0,u0±)∈Σ0± for some positive solutions u0± of (PB) with λ=0. It follows that Σ0±≠{(0,0)}, and by virtue of Corollary 4.9, that Σ0±∖{(0,0)} consists of nontrivial nonnegative solutions of (PB).
By definition, (λ,u)∈Σ0+ (resp. Σ0-) implies λ≥0 (resp. λ≤0). Lastly, by using Proposition 4.8 (ii) again, there exists ρ>0 small such that Σ0,ρ±:=Σ0±∩Bρ((0,0))¯ is closed and connected, and if (λ,u)∈Σ0,ρ±∖{(0,0)}, then λ≷0. So, 𝒞0±:=Σ0,ρ± have the desired properties. ∎
5.2 Proof of Assertion (ii) in Theorems 1.1 and 1.2
We consider the positivity of nontrivial nonnegative solutions of (PB) with f(s)=sq, q∈(0,1). Let 𝔖 be the nontrivial nonnegative solutions set of (PB), i.e.,
𝔖
:
=
{
(
λ
,
u
)
∈
ℝ
×
V
:
0
≢
u
≥
0
solves (
P
ℬ
)
}
.
Let ℭ be a nonempty connected subset of 𝔖, and let
ℭ
∘
:
=
{
(
λ
,
u
)
∈
ℭ
:
u
≫
0
}
.
The following lemma ([15, Theorem 1.7]) provides us with a nonexistence result for nontrivial nonnegative solutions of (1.12), which plays an important role in our argument when ℬu=∂u∂𝐧 and ∫Ωa≥0.
Lemma 5.3.
Let Bu=∂u∂n. Assume (Hψ) with ψ=a. If ∫Ωa≥0, then there exists qa*∈(0,1) such that (1.12) has no nontrivial nonnegative solutions for any q∈(qa*,1).
We give now sufficient conditions for the positivity of the nontrivial nonnegative solutions on ℭ as follows. We recall that the sets 𝒜ℬ±a are given by (1.13).
Proposition 5.4.
Let f(s)=sq, q∈(0,1). Suppose (1.14), (Hψ) with ψ=±a and the condition b≥0 in Ω. Assume additionally ∫Ωa<0 if Bu=∂u∂n. If
q
∈
𝒜
ℬ
a
∩
𝒜
ℬ
-
a
(Dirichlet),
q
∈
𝒜
ℬ
a
∩
(
q
-
a
∗
,
1
)
(Neumann),
where q-a∗ is as in Lemma 5.3, then C∘ is open and closed in C. Consequently, C∘=C if C∘≠∅.
Proof.
It is straightforward that ℭ∘ is open in ℭ, since u≫0 for (λ,u)∈ℭ∘. Next, we verify that ℭ∘ is closed in ℭ. Assume that (λn,un)∈ℭ∘ and (λn,un)→(λ0,u0)∈ℭ in ℝ×V. We shall show that (λ0,u0)∈ℭ∘. We discuss the following three cases, in accordance with the sign of λ0:
Case (i): λ0>0. We use the condition q∈𝒜ℬa to deduce the desired assertion. In this case, λn>0 for sufficiently large n. By the change of variables vn=λn-11-qun, we find that
{
-
Δ
v
n
=
a
(
x
)
v
n
q
+
λ
n
-
1
1
-
q
b
(
x
)
g
(
λ
n
1
1
-
q
v
n
)
in
Ω
,
ℬ
v
n
=
0
on
∂
Ω
.
Since vn→v0=λ0-11-qu0 in V, we find that v0 is a nonnegative weak solution of the problem
{
-
Δ
v
=
a
(
x
)
v
q
+
λ
0
-
1
1
-
q
b
(
x
)
g
(
λ
0
1
1
-
q
v
)
in
Ω
,
ℬ
v
=
0
on
∂
Ω
.
In addition, v0≢0 in Ω+a. Indeed, since (1.14) holds and b≥0, we see that vn is a supersolution of (1.12) which is positive in Ω. So, condition (Hψ) with ψ=a allows us to apply [15, Lemma 2.2], and deduce that there exist a ball B⋐Ω+a and a continuous function ψ on B¯ such that vn≥ψ>0 in B. Passing to the limit, we have that v0≥ψ in B, as desired.
By (1.14) and the condition b≥0, v0 is a supersolution of (1.12) and v0>0 in B. On the other hand, we can construct a nonnegative subsolution ψ0 of (1.12) such that ψ0≢0 in B, ψ0≡0 in Ω∖B and ψ0≤v0. The subsolutions and supersolutions method provides us with a solution v1 of (1.12) such that ψ0≤v1≤v0, so that v1≫0, since q∈𝒜ℬa. Consequently, we conclude that u0=λ01/(1-q)v0≫0, as desired.
Case (ii): λ0<0. (1) Case ℬu=u: We use the condition q∈𝒜ℬ-a to deduce the desired assertion. In this case, λn<0 for sufficiently large n. Setting μ=-λ, (PB) turns into
{
-
Δ
u
=
-
μ
a
(
x
)
u
q
+
b
(
x
)
g
(
u
)
in
Ω
,
u
=
0
on
∂
Ω
.
Set μn=-λn>0, so that vn:=μn-11-qun satisfies
{
-
Δ
v
n
=
-
a
(
x
)
v
n
q
+
μ
n
-
1
1
-
q
b
(
x
)
g
(
μ
n
1
1
-
q
v
n
)
in
Ω
,
v
n
=
0
on
∂
Ω
.
Since vn→v0=μ0-11-qu0 in C01(Ω¯), we infer that v0 is a nonnegative weak solution of the problem
{
-
Δ
v
=
-
a
(
x
)
v
q
+
μ
0
-
1
1
-
q
b
(
x
)
g
(
μ
0
1
1
-
q
v
)
in
Ω
,
v
=
0
on
∂
Ω
.
By using the condition q∈𝒜ℬ-a, the rest of the argument is carried out similarly to the previous case.
(2) Case ℬu=∂u∂𝐧: Under q∈(q-a∗,1), we shall see that case λ0<0 does not occur, using Lemma 5.3. We have λn<0 for n sufficiently large. By setting μ=-λ, (PB) becomes
{
-
Δ
u
=
-
μ
a
(
x
)
u
q
+
b
(
x
)
g
(
u
)
in
Ω
,
∂
u
∂
𝐧
=
0
on
∂
Ω
.
Setting μn=-λn>0 and vn=μn-11-qun, we find that
{
-
Δ
v
n
=
-
a
(
x
)
v
n
q
+
μ
n
-
1
1
-
q
b
(
x
)
g
(
μ
n
1
1
-
q
v
n
)
in
Ω
,
∂
v
n
∂
𝐧
=
0
on
∂
Ω
.
Since vn→v0=μ0-11-qu0 in C1(Ω¯), we infer that v0 is a nonnegative weak solution of the problem
{
-
Δ
v
=
-
a
(
x
)
v
q
+
μ
0
-
1
1
-
q
b
(
x
)
g
(
μ
0
1
1
-
q
v
)
in
Ω
,
∂
v
∂
𝐧
=
0
on
∂
Ω
.
In addition, v0≢0 in Ω+-a. Indeed, since (1.14) holds, and b≥0, we see that vn is a positive supersolution of
(5.5)
{
-
Δ
v
=
-
a
(
x
)
v
q
in
Ω
,
∂
v
∂
𝐧
=
0
on
∂
Ω
,
so that, by [15, Lemma 2.2], there exists a ball B⋐Ω+-a and a continuous function ψ on B¯ such that vn≥ψ>0 in B. Passing to the limit, we obtain v0≥ψ in B, as desired.
Now, we see that v0 is also a nonnegative supersolution of (5.5) such that v0>0 in B. Since we can construct a nonnegative subsolution ψ0 of (5.5) such that ψ0≢0 in B, ψ0≡0 in Ω∖B and ψ0≤v0, the subsolutions and supersolutions method provides us with a solution v1 of (5.5) such that ψ0≤v1≤v0 on Ω¯. So, v1 is nontrivial and nonnegative. However, this contradicts Lemma 5.3, since ∫Ω(-a)>0 and q-a*<q<1.
Case (iii): λ0=0. In this case, u0 solves the problem
{
-
Δ
u
0
=
b
(
x
)
g
(
u
0
)
in
Ω
,
ℬ
u
0
=
0
on
∂
Ω
.
Since u0 is nontrivial and nonnegative, the strong maximum principle and Hopf’s lemma yield that u0≫0, as desired.
Lastly, since ℭ is connected, we conclude that ℭ∘=ℭ if ℭ∘≠∅. ∎
Introducing the following growth condition on g:
(5.6)
0
<
lim
s
→
∞
g
(
s
)
s
p
<
∞
for some
p
>
1
,
where
p
<
N
+
1
N
-
1
if
N
>
2
,
we can deduce that ℭ∘≠∅, as shown in (i) and (ii) below.
Remark 5.5.
If (0,u0)∈ℭ with u0≢0, then u0≫0, i.e., (0,u0)∈ℭ∘. Indeed, this is a direct application of the strong maximum principle and Hopf’s lemma.
Assume (1.3) and (Hψ) with ψ=a. Assume also (1.11) if ℬu=u; and (5.6) and ∫Ωa<0 if ℬu=∂u∂𝐧. Let b≡1 and q∈𝒜ℬa. If there exist (λn,un)∈ℭ with λn→0+, then un≫0 for sufficiently large n, i.e., (λn,un)∈ℭ∘ for such n. This is an immediate consequence of [15, Theorem 4.1, Theorem 4.5].
When ℬu=∂u∂𝐧, Proposition 5.4 is valid for ∫Ωa>0, where we now assume q∈𝒜ℬ-a∩(qa*,1) instead of q∈𝒜ℬa∩(q-a*,1). Indeed, when ∫Ωa>0, the case λ0>0 does not occur, based on Lemma 5.3, whereas the case λ0<0 is verified as in the proof of Proposition 5.4, using λa(x)=(-λ)(-a(x)) and relying on [15, Lemma 2.2].
Proposition 5.4 and item (ii) hold more generally in the framework ℝ×C(Ω¯), which can been seen by using elliptic regularity.
Proof of assertion (ii) in Theorem 1.1.
We note from Remark 2.2 (iii) that (1.14) holds in case (b) of Theorem 1.1. Based on the result stated in Remark 5.5 (i), this assertion is verified by a direct application of Proposition 5.4. ∎
Proof of assertion (ii) in Theorem 1.2.
Based on the result stated in Remark 5.5 (ii), this assertion is straightforward from Proposition 5.4 and Remark 5.5 (iv). Indeed, we do not need to assume (Hψ) with ψ=-a for applying Proposition 5.4 to the loop 𝒞∗ given in Theorem 1.2 (ii), since it lies in λ≥0 (see Figure 1 (ii)). Note that the condition (Hψ) with ψ=-a is used only for case (ii) in the proof of Proposition 5.4. ∎
