Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity

Kazuhiro Ishige; Shinya Okabe; Tokushi Sato

doi:10.1515/anona-2021-0220

Open Access Published by De Gruyter February 25, 2022

Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity

Kazuhiro Ishige , Shinya Okabe and Tokushi Sato

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0220

Abstract

In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem

(P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞,

where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ^* > 0 such that problem (P) possesses a solution if 0 < κ < κ^* and it does not possess no solutions if κ > κ^*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ^*.

Keywords: inhomogeneous nonlinear elliptic equation; exponential nonlinearity

MSC 2010: 35B09; 35J61; 35B32

1 Introduction

This paper is concerned with the existence and the nonexistence of solutions to an inhomogeneous nonlinear elliptic problem

(P) {−Δu+u=F(u)+κμ in RN,u>0 in RN,u(x)→0 as |x|→∞,

where N ≥ 2, F ∈ C¹([0, ∞)) ∩ C²((0, ∞)), κ > 0, and μ∈Lc1(RN)\{0} is nonnegative. Throughout this paper we assume the following conditions on the nonlinear term F:

(Behavior of F as t → 0+)
F(0) = F′(0) = 0 and F″(t) = O(t^α−1) as t → +0 for some α ∈ (0, 1);
(Exponential nonlinearity of F)
F is a convex function in (0, ∞) such that

limt→∞F′(t)=∞, lim inft→∞F(t)F″(t)F′(t)2=1.

Conditions (F1) and (F2) are satisfied in the following cases:

F(t) = t^p exp(t^q) with p > 1 and q > 0;
F(t) = t^p exp(exp(t^q)) with p > 1 and q > 0.

In this paper, under a suitable integrability condition on μ, we prove the existence of the threshold parameter κ^* on the solvability of problem (P). Furthermore, in the case of 2 ≤ N ≤ 9, we show that problem (P) possesses a unique solution if κ = κ^*, and give a complete classification of the solvability of problem (P).

Let us recall some results on an inhomogeneous elliptic problem with power nonlinearity

(P’) {−Δu+u=up+κμ in RN,u>0 in RN,u(x)→0 as |x|→∞,

where N ≥ 2, κ > 0, and μ is a nonnegative Radon measure in R^N with compact support. The structure of solutions to problem (P’) has been studied in many papers. See e.g., [1,2,3,4,5,6,11,12,13,14,19,20,21,22,27,28,30,32,33,34,35], and references therein. Among others, under the assumption that

μ∈H−1(RN) and |x|N−2μ∈L∞(RN),

Deng and Li [11,12] proved the following assertions:

There exists κ^* > 0 such that problem (P’) possesses a solution if 0 < κ < κ^* and it possesses no solutions if κ > κ^*;
Let κ = κ^*. Then problem (P’) possesses a unique solution if 1 < p ≤ p_S;
Assume either 1 < p < p_S or p = p_S with 3 ≤ N ≤ 5. Then problem (P’) possesses at least two solutions in H¹(R^N) if 0 < κ < κ^*;
Let p = p_S and N ≥ 6. Under a suitable symmetric condition on μ, problem (P’) possesses a unique solution in H¹(R^N) for small enough κ > 0.

Here

pS=∞ if N=1,2, pS:=N+2N−2 if N≥3.

Subsequently, under more general conditions on μ, in [21], the authors of this paper obtained assertion (A) for problem (P’). Furthermore, we proved that assertion (B) holds in the Joseph-Lundgren subcritical case, that is,

1<p<pJL:={∞if N≤10,(N−2)2−4N+8N−1(N−2)(N−10)if N≥11.

In this paper, motivated by 21], we study the existence and the nonexistence of solutions to inhomogeneous elliptic problem (P) with exponential nonlinearity and obtain similar results to assertion (A). Furthermore, we prove that assertion (B) holds in the case of 2 ≤ N ≤ 9.

The structure of solutions to elliptic problems with exponential nonlinearity has also been studied in many papers. See e.g. [7,8,15,22,24,25,26] and references therein.

We introduce some notations and formulate a definition of solutions to problem (P). For x ∈ R^N and R > 0, let B(x, R) := {y ∈ R^N : |x − y| < R}. Define

C0(RN):={f∈C(RN) : lim|x|→∞f(x)=0},Lcr(RN):={f∈Lr(RN) : f has a compact support in RN },

where 1 ≤ r ≤ ∞. Similarly, we define Wc1,r(RN) , where 1 ≤ r ≤ ∞. We denote by G the fundamental solution of −Δv + v = 0 in R^N, that is,

(1.1) G(x):=1(2π)N/2|x|(N−2)/2K(N−2)/2(|x|), x∈RN\{0},

where K_(N−2)/2 is the modified Bessel function of order (N − 2)/2.

Definition 1.1

Let μ be a nonnegative measurable function in R^N and κ > 0.

We say that u is a solution to problem (P) if u∈C0(RN)+Lc∞(RN) and u satisfies
(1.2) u(x)=[G*F(u)](x)+κ[G*μ](x)>0 for almost all x∈RN.
We say that u is a supersolution to problem (P) if u∈C0(RN)+Lc∞(RN) and u satisfies
u(x)≥[G*F(u)](x)+κ[G*μ](x)>0 for almost all x∈RN.
Let u be a solution to problem (P). We say that u is a minimal solution to problem (P) if, for any solution v to problem (P), u(x) ≤ v(x) holds for almost all x ∈ R^N.

Now we are ready to state our results of this paper. Theorem 1.1 corresponds to assertion (A) and shows the existence of threshold parameter κ^* for the existence of solutions to problem (P).

Theorem 1.1

Let N ≥ 2 and F ∈ C¹([0, ∞)) ∩ C²((0, ∞)). Assume conditions (F1) and (F2). Let μ∈Lc1(RN)\{0} be nonnegative in R^N and

(1.3) G*μ∈L∞(RN).

Then there exists κ^* ∈ (0, ∞) with the following properties:

If 1 < κ < κ^*, then problem (P) possesses a minimal solution u^κ. Furthermore, u^κ(x) = O(G(x)) as |x| → ∞;
If κ > κ^*, then problem (P) possesses no solutions.

Theorem 1.2 corresponds to assertion (B) and shows the existence and the uniqueness of solutions to problem (P) with κ = κ^*. See also Theorem 6.1.

Theorem 1.2

Let N ≥ 2 and F ∈ C¹([0, ∞)) ∩ C²((0, ∞)). Assume conditions (F1) and (F2). Let μ∈Lc1(RN)\{0} be nonnegative in R^N and 2 ≤ N ≤ 9. Then problem (P) with κ = κ^* possesses a unique solution if either

lim supt→∞F′(t)F(t)<∞ and G * μ ∈ BC(R^N);
There exists μ¯∈Wc1,r(RN) with r = 4N/(N + 2) such that μ≤μ¯ in R^N.

We remark that G * μ ∈ BC(R^N) if μ∈Lcr for some r > N/2. (See (G2) in Section 2.)

Theorem 1.1 is proved by a modification of the proof of [21, Theorem 1.1]. For the proof of Theorem 1.2, we obtain uniform local L^∞-estimates of {u^κ} with respect to 0 < κ < κ^*, where u^κ and κ^* are as in Theorem 1.1. In this step, the arguments in [21] are not available since some key inequalities in [21] are not useful for exponential nonlinearity. Furthermore, it is not easy to introduce a general framework treating exponential nonlinearity. We prepare some inequalities peculiar to exponential nonlinearity under conditions (F1) and (F2), and obtain energy estimates of solutions to problem (P) to show

sup0<κ<κ*supz∈RN∥(1+F(uκ))1ν∥H1(B(z,1))<∞

with a suitable choice ν ∈ (1/2, 1) (see (6.1)). Then we apply the Sobolev imbedding theorem to obtain

sup0<κ<κ*supz∈RN∥F(uκ)∥LpS+1ν(B(z,1))<∞.

On the other hand, it follows from 2 ≤ N ≤ 9 that

limν→1/2pS+1ν=4NN−2>N2.

Then we apply elliptic regularity theorems with a suitable choice of ν to obtain uniform local boundedness and equi-continuity of {F(u^κ)} with respect to κ ∈ (0, κ^*). See Section 6.1. These imply that the limit function u^* := lim_{κ→κ^*−0} u^κ is bounded in R^N and it solves integral equation (1.2). Here lim_{κ→κ ^*−0} u^κ is the limit of u_κ as κ → κ^* with κ ∈ (0, κ^*). Next we prove that u^*(x) → 0 as |x| → ∞ by using the comparison between u^* and its mirror reflection with respect to hyperplanes (see Section 6.2). Finally we modify the arguments in [21, Section 6] to complete the proof of Theorem 1.2 (see Section 6.3).

The rest of this paper is organized as follows. In Section 2 we recall some properties of the fundamental solution and an eigenvalue problem to the elliptic operator −Δ+1. In Sections 3 and 4 we modify the arguments in [21] to obtain some estimates of approximate solutions and prove Theorem 1.1. In Section 5 we prove some inequalities related to the nonlinearity F, which play an important role in Section 6.1. In Section 6 we prove Theorem 1.2, and give a complete classification of the solvability of problem (P) in the case of 2 ≤ N ≤ 9.

2 Preliminaries

In this section we recall some properties on the fundamental solution G = G(x). In what follows, for any nonnegative functions f and g in R^N, we say that f(x) ≍ g(x) as x → 0 if there exists c > 0 such that c⁻¹g(x) ≤ f(x) ≤ cg(x) in a neighborhood of 0. Similarly, we say that f(x) ≍ g(x) as x → ∞ if there exists c > 0 such that c⁻¹g(x) ≤ f(x) ≤ cg(x) in a neighborhood of the space infinity. The letter C denotes generic positive constants and it may have different values also within the same line.

2.1 Fundamental solution G

We collect some properties of the fundamental solution G. It follows from (1.1) that G is radially symmetric in R^N and monotone decreasing with respect to |x|. Furthermore,

(2.1) G(x)≍{|x|−(N−2) if N≥3,−log|x| if N=2, as |x|→0,G(x)≍|x|−N−12e−|x| as |x|→∞.

We recall the following properties on the fundamental solution G. These follow from the Hölder inequality, the Hardy–Littlewood–Sobolev inequality, the Sobolev inequality, and elliptic regularity theorems (see e.g., [21, Section 2] and [27, Appendix]).

Let r > N/2. Then
G*v∈C0(RN) and ∥G*v∥L∞(RN)≤C∥v∥Lr(RN)
for v ∈ L^r(R^N).
For v ∈ L¹(R^N) ∩ L^r(R^N) with some r > N/2, then
G*v∈C0(RN)∩L1(RN)∩H1(RN).
Let v∈Lcr(RN) for some r > 1. Then
∇[G*v]∈LcNrN−r(RN) if 1<r<N and ∇[G*v]∈BC(RN) if r>N.

Set g := G * χ_B(0,1), where χ_B(0,1) is the characteristic function of the ball B(0, 1). Then

(2.2) g∈C∞, g(x)>0 in RN, g(x)≍G(x) as |x|→∞, |∇g|g∈BC(RN).

Furthermore, for any σ > 1, there exists C > 0 such that

(2.3) 0<[G*gσ](x)≤Cg(x) for x∈RN.

We define

gBC(RN):={f : g−1f∈BC(RN)},

which is a Banach space with the norm |||f||| := sup_x∈R^_N |g(x)⁻¹f(x)|.

2.2 Eigenvalue problem

Consider the eigenvalue problem

(2.4) −Δϕ+ϕ=λa(x)ϕ in RN, ϕ∈H1(RN),

where a ∈ L^N/2(R^N)∩L^r(R^N)\{0} for some r > N/2 and a(x) ≥ 0 for almost all x ∈ R^N. By the same arguments in the proofs of [27, Lemmas B2 and B3] we have the following lemma.

Lemma 2.1

Then problem (2.4) has the first eigenvalue λ₁ > 0 and the corresponding eigenfunction φ₁ with φ₁ > 0 in R^N. Furthermore,

λ1=inf{∥ψ∥H1(RN)2 / ∫RNaψ2 dx : ψ∈H1(RN), ∫RNaψ2 dx≠0}.

3 Approximate solutions

Let μ∈Lc1(RN)\{0} be nonnegative in R^N and satisfy (1.3). Then

μ0(x):=[G*μ](x)>0 for almost all x∈ RN , μ0∈C∞(RN\B(0,R)).

For any κ > 0, we define {Ujκ}j=0∞ and {Vjκ}j=0∞ by

(3.1) U0κ:=κμ0, Ujκ:=G*F(Uj−1κ)+κμ0, j=1,2,…,V0κ:=U0κ, Vjκ:=Ujκ−Uj−1κ, j=1,2,….

On the other hand, by (F1), for any M > 0, we find C_M > 0 such that

(3.2) 0≤F(t)≤CMtα+1, 0≤F′(t)≤CMtα, 0≤F″(t)≤CMtα−1,

for t ∈ (0, M]. Notice that α ∈ (0, 1).

Let u be a solution to problem (P). Let K > 0 and j ∈ {0, 1, . . . }. Then, by the same arguments as in [21, Section 3] we obtain the following properties:

(3.3) 0<C−1g(x)≤Ujκ(x)≤Uj+1κ(x)≤u(x),0≤Ujκ+ϵ(x)−Ujκ(x)≤CϵUjκ(x),0≤Vjκ+ϵ(x)−Vjκ(x)≤Cϵ,g−1Ujκ∈L∞(RN) and ∥g−1Ujκ∥L∞(RN)≤Cκ,g−1Vj+1κ∈BC(RN) and ∥g−1Vj+1κ∥BC(RN)≤Cκα(j+1)+1,

for x ∈ R^N, 0 < κ ≤ K, and ∊ ∈ (0, 1]. Furthermore, we have:

Lemma 3.1

Assume the same conditions as in Theorem 1.1. Let κ > 0. Then, for any j ∈ {0, 1, . . . }, there exists C > 0 such that

(3.4) 0≤Vjκ+ϵ(x)−Vjκ(x)≤Cϵαg(x), x∈RN,

for 0 < ∊ ≤ 1.

Proof

Let 0 < ∊ ≤ 1. Since V0κ+ϵ−V0κ=κ−1ϵU0κ , by (3.3) we have (3.4). Furthermore, by (F2), (3.1), (3.2), and (3.3) we see that

V1κ+ϵ−V1κ=G*[F(U0κ+ϵ)−F(U0κ)]≤G*[F′(U0κ+ϵ)(U0κ+ϵ−U0κ)]≤CϵG*[(U0κ)α+1]≤Cϵ[G*gα+1].

This together with (2.3) implies (3.4) with j = 1.

Assume that

(3.5) ∥g−1[Vjκ+ϵ−Vjκ]∥L∞(RN)≤Cϵα, 0<ϵ≤1,

for some j ∈ {1, 2, . . . }. It follows from (3.1) that

(3.6) Vj+1κ+ϵ(x)−Vj+1κ(x)=G*[F(Ujκ+ϵ)−F(Uj−1κ+ϵ)]−G*[F(Ujκ)−F(Uj−1κ)]=G*[h(x,1)−h(x,0)],

where

h(x,t):=F(α(x,t))−F(β(x,t)),α(x,t):=tUjκ+ϵ(x)+(1−t)Uj−1κ+ϵ(x), β(x,t):=tUjκ(x)+(1−t)Uj−1κ(x).

On the other hand, we have

(3.7) h′(x,t)=F′(α(x,t))Vjκ+ϵ−F′(β(x,t))Vjκ=F′(α(x,t))(Vjκ+ϵ−Vjκ)+(F′(α(x,t))−F′(β(x,t)))Vjκ

for 0 ≤ t ≤ 1. Furthermore, by (3.3) we see that

(3.8) 0≤α(x,t)−β(x,t)=t(Ujκ+ϵ(x)−Ujκ(x))+(1−t)(Uj−1κ+ϵ(x)−Uj−1κ(x))≤Cϵ(Ujκ(x)+Uj−1κ(x))≤Cϵg(x),max{α(x,t),β(x,t)}=α(x,t)≤Ujκ+ϵ(x)≤CUjκ(x)≤Cg(x),min{α(x,t),β(x,t)}=β(x,t)≥Uj−1κ(x)≥C−1g(x),

for x ∈ R^N, 0 ≤ t ≤ 1, and 0 < ∊ ≤ 1. Then, by (3.2), (3.5), and (3.8) we have

(3.9) 0≤F′(α(x,t))(Vjκ+ϵ(x)−Vjκ(x))≤Cϵαg(x)α(Vjκ+ϵ(x)−Vjκ(x))≤Cϵαg(x)α+1,

(3.10) (F′(α(x,t))−F′(β(x,t)))Vjκ(x)≤Cβ(x,t)α−1(α(x,t)−β(x,t))g(x)≤Cϵg(x)α+1,

for x ∈ R^N, 0 ≤ t ≤ 1, and 0 < ∊ ≤ 1. Combining (3.6), (3.7), (3.9), and (3.10), by (2.3) we obtain

0≤Vj+1κ+ϵ(x)−Vj+1κ(x)≤Cϵα[G*gα+1](x)≤Cϵαg(x)

for x ∈ R^N and 0 < ∊ ≤ 1. Therefore we see that (3.5) holds with j replaced by j + 1. Then, by induction we obtain the desired conclusion. Thus Lemma 3.1 follows.

4 Proof of Theorem 1.1

Let κ > 0. Assume the same conditions as in Theorem 1.1. By conditions (F1) and (F2) we apply the same arguments in [21, Section 4] to obtain the following properties.

The following two conditions are equivalent:
1. u=w+U1κ is a solution to problem (P);
2. w ∈ C₀(R^N) is positive in R^N and w satisfies
  (4.1) w=G*[F(w+U1κ)−F(U0κ)] in RN.
Let w satisfy (W1)-(b). Then 0 ≤ w(x) ≤ Cg(x) in R^N. Furthermore, w ∈ H¹(R^N) and w is a weak solution of
(4.2) −Δw+w=F+(w+U1κ)−F(U0κ) in RN,
that is,
(4.3) ∫RN[∇w⋅∇ψ+wψ] dx=∫RN[F+(w+U1κ)−F(U0κ)]ψ dx
for ψ ∈ H¹(R^N). Here F⁺(t) := F(max{t, 0}) for t ∈ R.
Let v be a supersolution to problem (P). Then Ujκ(x)≤v(x) in R^N for j ∈ {0, 1, . . . }. In addition, the limit function
U∞κ(x):=limj→∞Ujκ(x)
exists in R^N and it is a minimal solution to problem (P).

Furthermore, similarly to the proof of [21, Lemma 4.4], we have:

Lemma 4.1

Assume the same conditions as in Theorem 1.1. Set

t*:=sup{t≥0 : F(t)=0}.

Let κ_* ≥ 0 be such that t_* = κ_*‖G * μ‖_L^∞(R^N). For any κ > 0, set

Sκ:={u : u is a solution to problem (P) }.

Then the following properties hold.

There exists ∊ > 0 such that S_κ ≠ ∅ for κ ∈ (0, κ_* + ∊).
If S_κ ≠ ∅ for some κ > 0, then problem (P) possesses a minimal solution u^κ.

Notice that t_* < ∞ and κ_* < ∞ under the assumptions of Lemma 4.1.

Proof

We prove assertion (i). If 0 < κ ≤ κ_*, then F(U0κ)=F(κG*μ)=0 in R^N. This together with (3.3) implies that uκ=U0κ in R^N. So it suffices to consider the case of κ > κ_*.

Let κ_* < κ < κ_* + 1, and define

Tκ[v]:=G*[F+(v+U1κ)−F(U0κ)] for v∈gBC(RN).

We prove that T_κ with κ_* < κ < κ_* + ∊ is a contraction mapping on

BgBC(0,δ)¯:={f∈gBC(RN) : |||f|||≤δ}

for some δ > 0 and ∊ ∈ (0, 1). It follows from (F1) and (F2) that

t*=sup{t≥0 : F′(t)=0}=sup{t≥0 : F″(t)=0}.

Then, for any L > 0, by (F1) we see that

(4.4) 0≤F′(t)=∫0tF″(τ) dτ=∫stF″(τ) dτ≤C∫stτα−1 dτ≤C∫st(τ−s)α−1 dτ≤C(t−s)α

for 0 ≤ s < t < t_* + L with t_* ≥ s. Since 0≤U1κ*=U0κ*≤t* and V1κ*=0 in R^N, by Lemma 3.1, (3.3), and (4.4) we have

|F+(v+U1κ)−F(U0κ)|≤F′(|v|+U1κ)(|v|+V1κ)=F′(|v|+U1κ)(|v|+V1κ−V1κ*)≤C(|v|+U1κ−U1κ*)α(|v|+|V1κ−V1κ*|)≤C(|||v|||+|||U1κ−U1κ*|||)α(|||v|||+|||V1κ−V1κ*|||)gα+1≤C(|||v|||+κ−κ*)α(|||v|||+(κ−κ*)α)gα+1≤C(|||v|||+(κ−κ*)α)α+1gα+1

for v ∈ gBC(R^N) with |||v||| ≤ 1. This together with (2.3) implies that

(4.5) |||Tκ[v]|||≤C(|||v|||+(κ−κ*)α)α+1

for v ∈ gBC(R^N) with |||v||| ≤ 1. Similarly, by (3.3) and (4.4) we have

|F+(v+U1κ)−F+(v˜+U1κ)|≤|F′(|v|+|v˜|+U1κ)||v−v˜|≤C(|v|+|v˜|+U1κ−U1κ*)α|v−v˜|≤C(|||v|||+|||v˜|||+|||U1κ−U1κ*|||)α|||v−v˜|||gα+1≤C(|||v|||+|||v˜|||+κ−κ*)α|||v−v˜|||gα+1

for v ∈ gBC(R^N) with |||v||| ≤ 1. This together with (2.3) implies that

(4.6) |||Tκ[v]−Tκ[v˜]|||≤C(|||v|||+|||v˜|||+κ−κ*)α|||v−v˜|||

for v, v˜∈gBC(RN) with |||v|||, |||v˜|||≤1 .

Let δ ∈ (0, 1) and ∊ ∈ (0, 1) be small enough. By (4.5) and (4.6) we have

BgBC(0,δ)¯∋v→Tκ[v]∈BgBC(0,δ)¯,|||Tκ[v]−Tκ[v˜]|||≤12|||v−v˜||| for v,v˜∈BgBC(0,δ)¯,

for κ_* < κ < κ_* + ∊. This means that T_κ is a contraction mapping on BgBC(0,δ)¯ .

Define a sequence {w_k} ⊂ gBC(R^N) by

w0:=0, wk:=Tκ[wk−1] (k=1,2,…).

Due to (3.3), by induction we see that 0 < w₁ < w₂ < · · · < w_k < · · · in R^N. Applying the fixed point theorem, we can find a positive function w_∞ ∈ gBC(R^N) such that

limk→∞|||w∞−wk|||=0, w∞=Tκ[w∞].

Furthermore, we see that w_∞ satisfies (4.1). Then we deduce that u=U1κ+w∞ is a solution to problem (P), and obtain assertion (i). Assertion (ii) follows from property (W3). Thus the proof of Lemma 4.1 is complete.

Let κ > κ_* and u^κ be a minimal solution to problem (P). Then u^κ > u^κ_* and F′(u^κ) ≢ 0 in R^N. Consider the linearized eigenvalue problem to problem (P) at u^κ

(E_κ) −Δϕ+ϕ=λF′(uκ)ϕ in RN, ϕ∈H1(RN).

By Lemma 2.1 we see that problem (E_κ) has the first eigenvalue λ1κ>0 and

(4.7) λ1κ=inf{∥ψ∥H1(RN)2 / ∫RNF′(uκ)ψ2 dx : ψ∈H1(RN), ∫RNF′(uκ)ψ2 dx≠0}.

Let ϕ1κ be the first eigenfunction to problem (E_κ) such that ϕ1κ>0 in R^N. By the same arguments as in the proof of [21, Lemma 4.5] we see that

(4.8) ϕ1κ(x)=λ1κ∫RNG(x−y)F′(uκ(y))ϕ1κ(y) dy,

(4.9) c−1g(x)≤ϕ1κ(x)≤cg(x),

for x ∈ R^N. Furthermore, we have:

Lemma 4.2

Assume the same conditions as in Theorem 1.1. Define

κ*:=sup{κ>0 : Sκ≠∅}.

For any κ_* < κ < κ^*, let λ1κ be the first eigenvalue to problem (E_κ). Then λ1κ>1 and

(4.10) ∫RN[|∇ψ|2+ψ2] dx≥λ1κ∫RNF′(uκ)ψ2 dx>∫RNF′(uκ)ψ2 dx

for ψ ∈ H¹(R^N) if

∫RNF′(uκ)ψ2 dx≠0.

Furthermore,

(4.11) 1<λ1κ′≤λ1κ if κ*<κ≤κ′<κ*.

Proof

Let κ_* < κ < κ′ < κ^*. Set wκ:=uκ−U1κ , wκ′:=uκ′−U1κ′ , and ψ := w^κ′ − w^κ. It follows from (3.3) and property (W3) that

(4.12) wκ′=limj→∞[Ujκ′−U1κ′]=limj→∞[Vjκ′+Vj−1κ′+…+V2κ′]>limj→∞[Vjκ+Vj−1κ+…+V2κ]=limj→∞[Ujκ−U1κ]=wκ.

Since the function [t_*, ∞) ∋ s → F(t + s) − F(s) is strictly monotone increasing for any fixed t > 0, by (3.3) we see that

(4.13) F(wκ′+U1κ′)−F(U0κ′)=F(wκ′+V1κ′+U0κ′)−F(U0κ′) >F(wκ′+V1κ′+U0κ)−F(U0κ) ≥F(wκ′+V1κ+U0κ)−F(U0κ)=F(wκ′+U1κ)−F(U0κ)≥0

for x∈E:={x∈RN : U0κ>t*} . It follows from κ > κ_* that |E| > 0. Similarly, since the function [0, ∞) ∋ s → F(t + s) − F(s) is monotone increasing for any fixed t > 0, by (3.3) we have

(4.14) F(wκ′+U1κ′)−F(U0κ′)=F(wκ′+V1κ′+U0κ′)−F(U0κ′) ≥F(wκ′+V1κ′+U0κ)−F(U0κ) ≥F(wκ′+V1κ+U0κ)−F(U0κ)=F(wκ′+U1κ)−F(U0κ)

for x ∈ R^N \ E. Since ϕ1κ is the eigenfunction of (E_κ) and positive, by (4.12) we have

(4.15) λ1κ∫RNF′(uκ)ψϕ1κ dx>0.

Since |E| > 0, by (4.3), (4.13), and (4.14) we obtain

(4.16) λ1κ∫RNF′(uκ)ψϕ1κ dx=∫RN[∇ψ∇ϕ1κ+ψϕ1κ] dx=∫RN[F(wκ′+U1κ′)−F(U0κ′)−F(wκ+U1κ)+F(U0κ)]ϕ1κ dx>∫RN[F(wκ′+U1κ)−F(wκ+U1κ)]ϕ1κ dx≥∫RNF′(uκ)ψϕ1κ dx.

Combining (4.15) with (4.16), we see that λ1κ>1 . This together with (4.7) implies (4.10) and (4.11). Thus Lemma 4.2 follows.

Now we are ready to complete the proof of Theorem 1.1.

Proof of Theorem 1.1

We show that κ^* < ∞. Consider the eigenvalue problem

(4.17) −Δψ+ψ=λψ in B(0,1), ψ=0 on ∂B(0,1), ψ∈H01(B(0,1)).

Let λ_B and ψ_B be the first eigenvalue and the first eigenfunction to problem (4.17), respectively, such that ψ_B > 0 in B(0, 1). Set ψ_B = 0 outside B(0, 1). Then ψ_B ∈ H¹(R^N) and

(4.18) ∫RN[|∇ψB|2+ψB2] dx=∫B(0,1)[|∇ψB|2+ψB2] dx=λB∫B(0,1)ψB2 dx>0.

On the other hand, it follows from Lemma 4.2, (F2), and (3.3) that

(4.19) ∫RN[|∇ψB|2+ψB2] dx>∫RNF′(uκ)ψB2 dx≥∫RNF′(U0κ)ψB2 dx≥∫RNF′(C−1κg(x))ψB2 dx≥C−1F′(C−1κminx∈B¯(0,1)g(x))

for κ_* < κ < κ^*. Since F′(t) → ∞ as t → ∞, by (4.18) and (4.19) we see that κ^* < ∞. This together with Lemma 4.1 implies assertions (i) and (ii) of Theorem 1.1. Thus Theorem 1.1 follows.

5 Some inequalities related to (F1) and (F2)

In this section we obtain some inequalities related to (F1) and (F2), which are used in Section 6.1. The following two lemmas concern with the growing up rate of F(t) and F′(t) as t → ∞.

Lemma 5.1

Assume the same conditions as in Theorem 1.1. Then

limt→∞F(t)tq=limt→∞F′(t)tq=∞ for q>0.

Proof

Let q > 0. Let ∊ > 0 be such that 1/∊ > q + 1. By (F2) we find T_∊ > 0 such that

(logF′(t))′≥(1−ϵ)(logF(t))′, t≥Tϵ.

This implies that

logF′(t)≥(1−ϵ)logF(t)−C, t≥Tϵ,

that is,

F′(t)≥e−CF(t)1−ϵ, t≥Tϵ.

Then

F(t)≥[F(Tϵ)ϵ+ϵe−C(t−Tϵ)]1ϵ, t≥Tϵ.

On the other hand, by (F1) and (F2) we have

F(t)=F(t)−F(0)≤tF′(t), t≥0.

Therefore we deduce that

limt→∞F(t)tq≥limt→∞F(t)tq+1=∞, limt→∞F′(t)tq=limt→∞F(t)tq+1tF′(t)F(t)=∞.

The proof is complete.

Lemma 5.2

Assume the same conditions as in Theorem 1.1. Set

(5.1) γ(t):=F′(t)1+F(t) for t≥0.

Then F(t)γ(t) ≤ F′(t) for t ≥ 0. Furthermore,

(5.2) lim inft→∞γ′(t)γ(t)2=0, limt→∞γ(t)t=∞.

Proof

It easily follows from (5.1) that F(t)γ(t) ≤ F′(t) for t ≥ 0. Furthermore, by (F2) we have

lim inft→∞γ′(t)γ(t)2=lim inft→∞F″(t)(1+F(t))−F′(t)2(1+F(t))2(1+F(t))2F′(t)2=lim inft→∞[F″(t)(1+F(t))F′(t)2−1]=0.

We prove that lim_t→∞ γ(t)t = ∞. Set

η(t):=γ(t)t=tF′(t)1+F(t).

It follows that

(5.3) η′(t)=(F′(t)+tF″(t))(1+F(t))−tF′(t)2(1+F(t))2=F′(t)1+F(t)+tF″(t)(1+F(t))−F′(t)2(1+F(t))2=t−1η(t)+t−1η(t)2[F″(t)(1+F(t))F′(t)2−1], t>0.

Then we have

(5.4) lim supt→∞ η(t)=∞.

Indeed, if not, then L := sup_t>0 η(t) < ∞. Let 0 < ∊ < 1/2L. Then, by (F2) and (5.3) we find T > 0 such that

(5.5) η′(t)≥t−1η(t)−ϵt−1η(t)2≥12t−1η(t), t≥T.

This implies that

logη(t)−logη(T)≥12logtT, t≥T.

Then we see that η(t) → ∞ as t → ∞, which is a contradiction. Thus (5.4) holds.

Let L′ > 0. Assume that lim inf_t→∞ η(t) < L′. Then, by (5.4) we find a sequence {t_j} ⊂ (0, ∞) such that

(5.6) limj→∞tj=∞, η(tj)≤L′, η′(tj)≤0.

Similarly to (5.5), by (5.3) we obtain

η′(tj)≥tj−1η(tj)−ϵtj−1η(tj)2≥12tj−1η(tj)>0 with ϵ=12L′.

This contradicts (5.6). So we see that lim inf_t→∞ η(t) ≥ L′. Since L′ is arbitrary, we obtain lim_t→∞ η(t) = ∞. Thus (5.2) holds, and Lemma 5.2 follows.

Let t_* ≥ 0 be as in Lemma 4.1. Then it follows from (F1) and (F2) that

t*=sup{t≥0 : F′(t)=0}=sup{t≥0 : γ(t)=0}.

For ν ∈ (0, 1), set

(5.7) Hν(t):=2ν∫0t(1+F(τ))2νγ(τ)2 dτ, hν(t):={Hν(t)γ(t) for t>t*,0 for 0≤t≤t*.

By (F1) and (F2) we find δ > 0 such that

Hν(t)=2ν∫t*t(1+F(τ))2νγ(τ)2 dτ≤C∫t*tF′(τ)2 dτ≤CF′(t)2(t−t*)≤Cγ(t)2(t−t*)

for t ∈ [t_*, t_* + δ). Then we see that h_ν is continuous in [0, ∞) and H_ν(t) = γ(t)h_ν(t) for t ≥ 0. In the following lemmas we obtain some inequalities on H_ν and h_ν.

Lemma 5.3

Assume the same conditions as in Theorem 1.1. Let 0 < ν < 1 and 0 < δ ≤ 1. Then there exists C > 0 such that

(1+F(t))2ν≤δHν(t)t+C for t≥0.

Proof

Let 0 < ν < 1. It follows from (F2) that

(1+F(t))2νγ(t)2=(1+F(t))2ν−2F′(t)2→∞ as t→∞,

which implies that lim_t→∞ H_ν(t) = ∞. On the other hand, by Lemma 5.1 we see that lim_t→∞ F(t)^2/ν/t = ∞. Since

[t−1(1+F(t))2ν]′=t−1(1+F(t))2ν[−t−1+2ν(1+F(t))−1F′(t)]=t−1(1+F(t))2νγ(t)[−1γ(t)t+2ν],

it follows from L’Hospital's rule and (5.2) that

limt→∞Hν(t)t(1+F(t))2ν=limt→∞2ν(1+F(t))2νγ(t)2/[t−1(1+F(t))2ν]′=limt→∞2νγ(t)t/[−1γ(t)t+2ν]=∞.

Then we obtain the desired conclusion, and the proof is complete.

Lemma 5.4

Assume the same conditions as in Theorem 1.1. Let 0 < ν < 1 and 0 < δ ≤ 1. Then there exists C > 0 such that

(5.8) hν(t)≤(1+δ)(1+F(t))2ν+C for t≥0.

Furthermore,

hν(t)≤22−ν(1+F(t))2ν for t≥0 ,
(2ν−δ)(1+F(t))2ν−νhν(t)≥0 for t≥0 if 0<δ≤2ν(1−ν)2−ν .

Proof

Let ∊ ∈ (0, 1]. By (5.2) we find t_∊ ≥ 0 such that

(5.9) −γ′(t)≤ϵγ(t)2 for t≥tϵ.

Then, combining (5.9) with the definition of γ, we have

2ν∫tϵt(1+F(τ))2νγ(τ)2 dτ=∫tϵt[ddτ(1+F(τ))2ν]γ(τ) dτ=[(1+F(τ))2νγ(τ)]τ=tϵτ=t−∫tϵt(1+F(τ))2νγ′(τ) dτ≤[(1+F(τ))2νγ(τ)]τ=tϵτ=t+ϵ∫tϵt(1+F(τ))2νγ(τ)2 dτ

for t ≥ t_∊. This together with (5.7) implies that

(5.10) (1−νϵ2)(Hν(t)−Hν(tϵ))≤(1+F(t))2νγ(t)−(1+F(tϵ))2νγ(tϵ)

for t ≥ t_∊. Then we deduce from Lemma 5.1, (5.2), and (5.7) that

(1−νϵ2)lim supt→∞hν(t)(1+F(t))2ν≤1.

Since ∊ is arbitrary, we deduce that

lim supt→∞hν(t)(1+F(t))2ν≤1,

which implies (5.8). On the other hand, by the definition of γ and (F2) we have

γ(t)2+γ′(t)=F″(t)1+F(t)≥0.

This means that inequality (5.9) holds with ∊ = 1 and t_∊ = 0. Therefore, by (5.10) with ∊ = 1 and t_∊ = 0 we see that

(1−ν2)Hν(t)≤(1+F(t))2νγ(t)

for t ≥ 0. This implies assertion (1). Furthermore, it follows from assertion (1) that

(2ν−ϵ)(1+F(t))2ν−νhν(t)≥(2ν−ϵ−2ν2−ν)(1+F(t))2ν≥0

for t ≥ 0 if

2ν−ϵ−2ν2−ν≥0.

Setting δ = ∊, we obtain assertion (2). Thus Lemma 5.4 follows.

Lemma 5.5

Assume the same conditions as in Theorem 1.1. Assume that

(5.11) lim supt→∞F′(t)F(t)<∞.

Then, for any ν ∈ (0, 1), δ ∈ (0, 1), and M > 0, there exists C > 0 such that

(5.12) Hν(t)≤γ(s+t)[(1+δ)(1+F(t))2ν+C]

for s ∈ [0, M] and t ≥ 0.

Proof

Let ∊ ∈ (0, 1] and t_∊ ≥ 0 be as in (5.9). It follows from (5.11) that

γ¯:=supt≥0γ(t)<∞.

Then, by (5.9) we have

ddt[log(1+F(τ))]τ=tτ=t+ϵ′=γ(t+ϵ′)−γ(t)=∫tt+ϵ′γ′(τ) dτ≥−ϵ∫tt+ϵ′γ(τ)2 dτ≥−ϵγ¯∫tt+ϵ′γ(τ) dτ=−ϵγ¯[log(1+F(τ))]τ=tτ=t+ϵ′

for t ≥ t_∊ and ∊′ > 0. This implies that the function

[tϵ,∞)∋t↦eϵγ¯t[log(1+F(t+ϵ′))−log(1+F(t))]

is monotone increasing. Then we obtain

log(1+F(t+ϵ′))−log(1+F(t))ϵ′≤eϵγ¯slog(1+F(t+s+ϵ′))−log(1+F(t+s))ϵ′

for t ≥ t_∊, s ≥ 0, and ∊′ > 0. Letting ∊′ → +0, we see that

(5.13) γ(t)≤eϵγ¯sγ(t+s)

for t ≥ t_∊, and s ≥ 0.

Let M > 0 and 0 < δ′ < δ. By (5.8) and (5.13), taking small enough ∊ ∈ (0, 1), we have

(5.14) Hν(t)≤eϵγ¯Mγ(t+s)[(1+δ′)(1+F(t))2ν+C]≤γ(t+s)[(1+δ)(1+F(t))2ν+C]

for t ≥ t_∊ and s ∈ [0, M]. On the other hand, by the convexity of F and (5.8) we see that

(5.15) Hν(t)≤F′(t)1+F(t+s)1+F(t+s)1+F(t)[(1+δ)(1+F(t))2ν+C]≤Cγ(t+s)[(1+δ)(1+F(t))2ν+C]

for 0 ≤ t ≤ t_∊ and s ∈ [0, M]. We combine (5.14) and (5.15) to obtain (5.12). Thus Lemma 5.5 follows.

6 Proof of Theorem 1.2

For 0 < κ < κ^*, let u^κ be the minimal solution to problem (P). Set

wκ:=uκ−U1κ.

In Section 6.1 we obtain uniform L^∞-local estimates of {w^κ} with respect to 0 < κ < κ^* by using inequalities given in Section 5. Then we see that u^* := lim_κ→κ^_*₋₀ u^κ is bounded and it solves integral equation (1.2). In Section 6.2 we prove that u^*(x) → 0 as |x| → ∞. In Section 6.3 we complete the proof of Theorem 1.2.

6.1 Uniform estimates of w^κ

The main purpose of this subsection is to prove the following proposition.

Proposition 6.1

Let 2 ≤ N ≤ 9, and assume the same conditions as in Theorem 1.1. Further assume either

lim supt→∞F′(t)F(t)<∞ or
There exists μ¯∈Wc1,r(RN) with r = 4N/(N + 2) such that μ≤μ¯ in R^N.

Then {w^κ} are uniformly bounded in R^N and equi-continuous for compact sets in R^N with respect to 0 < κ < κ^*. Here κ^* is as in Theorem 1.1.

For this aim we prove the following lemma by the use of Lemma 5.4.

Lemma 6.1

Let ν ∈ (1/2, 1) be such that

(6.1) 2ν−1≤2ν(1−ν)2−ν.

Then, for any δ ∈ (0, 2ν − 1], there exists C > 0 such that

(2ν−δ)∥[(1+F(v)gz]1ν∥H1(RN)2≤(v,Hν(v)gz2ν)H1(RN)+C

for nonnegative functions v ∈ H¹(R^N) and z ∈ R^N. Here g_z(x) := g(x − z) for x, z ∈ R^N.

Proof

Let ν ∈ (1/2, 1) be as in (6.1) and z ∈ R^N. Let v ∈ H¹(R^N) be nonnegative and set

Ψ(x):=[(1+F(v(x)))gz(x)]1ν, x∈RN.

Since

∇Ψ=1ν[(1+F(v))gz]1ν−1[F′(v)gz∇v+(1+F(v))∇gz]=1ν[(1+F(v))gz]1ν[γ(v)∇v+∇gzgz],

we have

(6.2) |∇Ψ|2=1ν2[(1+F(v))gz]2ν[γ(v)2|∇v|2+2γ(v)∇v⋅∇gzgz+|∇gz|2gz2].

On the other hand, it follows that

(6.3) ∇[Hν(v)gz2ν]=2νgz(x)2ν[(1+F(v))2γ(v)2∇v+Hν(v)∇gzgz].

By (6.2) and (6.3) we have

(6.4) (2ν−δ)∥[(1+F(v))gz]1ν∥H1(RN)2−(v,Hν(v)gz2ν)H1(RN)=−1ν2∫RN[δ(1+F(v))2νγ(v)2|∇v|2+ν2Hν(v)v]gz2ν dx +2ν2∫RN[(2ν−δ)(1+F(v))2νγ(v)−νHν(v)]∇v⋅∇gzgzgz2ν dx +2ν−δν2∫RN(1+F(v))2ν[ν2+|∇gz|2gz2]gz2ν dx.

Let 0 < δ ≤ 2ν − 1. Then it follows from (6.1) that

0<δ<2ν−1≤2ν(1−ν)2−ν.

By Lemma 5.4 we see that

(2ν−δ)(1+F(v))2ν−νhν(v)≥0.

This together with (2.2) and (6.4) implies that

(2ν−δ)∥[(1+F(v)gz]1ν∥H1(RN)2−(v,Hν(v)gz2ν)H1(RN)≤−1ν2∫RN[δ(1+F(v))2νγ(v)2|∇v|2+ν2Hν(v)v]gz2ν dx +2Cν2∫RN(2ν−δ)(1+F(v))2νγ(v)|∇v|gz2ν dx +C∫RN(1+F(v))2νgz2ν dx≤−∫RNHν(v)vgz2ν dx+C∫RN(1+F(v))2νgz2ν dx.

Then, by Lemma 5.3, (2.1), and (2.2) we obtain

(2ν−δ)∥[(1+F(v))gz]1ν∥H1(RN)2−(v,Hν(v)gz2ν)H1(RN)≤C.

Then Lemma 6.1 follows.

We prove Proposition 6.1 under condition (i).

Proof of Proposition 6.1 under condition (i)

Since w^κ = 0 if 0 < κ ≤ κ_*, it suffices to consider the case of κ_* < κ < κ^*.

Assume condition (i). Let ν ∈ (1/2, 1) be as in (6.1) and 0 < δ < (2ν − 1)/2. It follows from (4.2) that

(wκ,Hν(wκ)gz2ν)H1(RN)=∫RN[F(wκ+U1κ)−F(U0κ)]Hν(wκ)gz2ν dx≤∫RNF(wκ+U1κ)Hν(wκ)gz2ν dx

for z ∈ R^N and κ_* < κ < κ^*. Then, by Lemma 5.5 with M=∥U1κ*∥∞ and (5.1) we obtain

(wκ,Hν(wκ)gz2ν)H1(RN)≤∫RNF(uκ)γ(uκ)[(1+δ)(1+F(wκ))2ν+C]gz2ν dx≤(1+δ)∫RNF′(uκ)[(1+F(wκ))gz]2ν dx+C∫RNF′(uκ)gz2ν dx.

Applying Lemmas 4.2 and 6.1, we obtain

(2ν−δ)∥[(1+F(wκ))gz]1ν∥H1(RN)2−C≤(wκ,Hν(wκ)gz2ν)H1(RN)≤(1+δ)∥[(1+F(wκ))gz]1ν∥H1(RN)2+C∥gz1ν∥H1(RN)2

for z ∈ R^N and κ_* < κ < κ^*. Since 2ν − δ > 1 + δ, we deduce that

(6.5) supκ*<κ<κ*supz∈RN∥[(1+F(wκ))gz]1ν∥H1(RN)2<∞.

Consider the case when 3 ≤ N ≤ 9. Since

pS:=N+2N−2>32,

we can assume, without loss of generality, that

12<ν<pS−1=4N−2,

which implies that

(6.6) pS+1ν=1ν2NN−2>N2.

Applying the Sobolev inequality, by (6.5) we have

sup0<κ<κ*supz∈RN∥[(1+F(wκ))gz]1ν∥LpS+1(RN)<∞,

which implies that

(6.7) sup0<κ<κ*supz∈RN∥F(wκ)∥LpS+1ν(B(z,1))<∞.

On the other hand, it follows from condition (i) that F(s + t) ≤ e^CsF(t) + C for t ∈ [0, ∞) and s ≥ 0. This implies that

(6.8) F(uκ)≤F(∥U1κ*∥∞+wκ)≤CF(wκ)+C in RN

for κ_* < κ < κ^*. Combining (6.7) and (6.8), we obtain

(6.9) sup0<κ<κ*supz∈RN∥F(uκ)∥LpS+1ν(B(z,1))<∞.

Recalling uκ=wκ+U1κ and U0κ∈L∞(RN) , by (6.6) we apply the elliptic regularity theorems (see e.g. [31, Theorems 1 and 7]) to elliptic equation (4.2). Then we see that {w^κ}_{κ_*<κ<κ_*} are uniformly bounded in R^N and equi-continuous for compact sets in R^N. Thus Proposition 6.1 follows in the case of 3 ≤ N ≤ 9 under condition (i).

Similarly, in the case of N = 2, by the Sobolev inequality and (6.5) we see that

supκ*<κ<κ*supz∈RN∥F(uκ)∥Lr(B(z,1))<∞

for r ∈ [1, ∞), instead of (6.9). Then, applying the elliptic regularity theorems again, we see that {w^κ}_{κ_*<κ<κ}^_* are uniformly bounded in R^N and equi-continuous for compact sets in R^N. Thus Proposition 6.1 follows in the case of N = 2 under condition (i). The proof of Proposition 6.1 is complete under condition (i).

We prepare the following lemma for the proof of Proposition 6.1 under condition (ii).

Lemma 6.2

Let 2 ≤ N ≤ 9 and μ be a nonnegative measurable function in R^N with compact support. Let μ¯∈Wc1,r(RN) with r = 4N/(N + 2) such that 0≤μ(x)≤μ¯(x) for almost all x ∈ R^N. Then

G*μ∈H1(RN)∩C0(RN), μ¯1ν∈Hc1(RN) for 12<ν<1.

Proof

Let 2 ≤ N ≤ 9 and μ¯∈Wc1,r(RN) with r = 4N/(N + 2). Then r = 2 if N = 2 and r < N if N ≥ 3. Let 1/2 < ν < 1. The Sobolev imbedding theorem implies that

(6.10) μ¯1ν∈Lc2(RN) if N=2, μ¯∈LcNrN−r(RN) if N≥3.

Since

2ν<4<NrN−r with r=4NN+2,

we have μ¯1/ν∈Lc2(RN) if N ≥ 3. So we see that μ¯1/ν∈Lc2(RN) .

On the other hand, it follows from r = 4N/(N + 2) and 1/2 < ν < 1 that

NN−r=2r−2>1r−2(2ν−2).

This implies that

(6.11) NrN−r/(2ν−2)>rr−2=(r2)′.

Here (r/2)^′ is the Hölder conjugate of r/2 ≥ 1. Since |∇μ¯|2∈Lcr/2(RN) and μ¯∈LcNrN−r(RN) , by (6.11) we have

|∇μ¯1ν|2=1ν2μ¯2ν−2|∇μ¯|2∈Lc1(RN).

Combining μ¯1/ν∈Lc2(RN) , we see that μ¯1/ν∈Hc1(RN) .

On the other hand, it follows from N ≤ 9 and r = 4N/(N + 2) that

NrN−r>N2 if N≥3.

Then it follows from (G2), (6.10), and 0≤μ≤μ¯ in R^N that G * μ ∈ C₀(R^N) ∩ H¹(R^N). Thus Lemma 6.2 follows.

Proof of Proposition 6.1 under condition (ii)

Similarly to the case of condition (i), it suffices to consider the case when κ_* < κ < κ^*. Assume condition (ii). Let ν ∈ (1/2, 1) be as in (6.1) and 0 < δ < (2ν − 1)/2. It follows from (P), (5.1), and (5.7) that

(uκ,Hν(uκ)gz2ν)H1(RN)=∫RN(F(uκ)+κμ)Hν(uκ)gz2ν dx≤∫RNF′(uκ)hν(uκ)gz2ν dx+κ∫RNμ¯γ(uκ)hν(uκ)gz2ν dx

for z ∈ R^N and κ_* < κ < κ^*. This together with Lemmas 5.4 and 6.1 implies that

(2ν−δ)∥[(1+F(uκ))gz]1ν∥H1(RN)2−C≤(uκ,Hν(uκ)gz2ν)H1(RN)≤∫RNF′(uκ)hν(uκ)gz2ν dx+2κ2−ν∫RNμ¯γ(uκ)(1+F(uκ))2νgz2ν dx≤∫RNF′(uκ)[(1+δ)(1+F(uκ))2ν+C]gz2ν dx+2κ2−ν∫RNμ¯F′(uκ)(1+F(uκ))2ν−1gz2ν dx

for z ∈ R^N and κ_* < κ < κ^*. By Lemma 4.2 and the Hölder inequality we have

(2ν−δ)∥[(1+F(v))gz]1ν∥H1(RN)2−C≤(1+δ)∥[(1+F(v))gz]1ν∥H1(RN)2+C∥gz1ν∥H1(RN)2 +2κ2−ν(∫RNμ¯(x)2νF′(uκ)gz2ν dx)ν2(∫RNF′(uκ)(1+F(uκ))2νgz2ν dx)2−ν2≤(1+δ)‖[(1+F(v))gz]1ν‖H1(RN)2+C+2κ2−ν∥μ¯1νgz1ν∥H1(RN)ν∥[(1+F(v))gz]1ν∥H1(RN)2−ν

for z ∈ R^N and κ_* < κ < κ^*. Since 2ν − δ > 1 + δ, we deduce that

supκ*<κ<κ*supz∈RN∥[(1+F(uκ))gz]1ν∥H1(RN)2<∞.

Then, similarly to the proof under condition (i), the Sobolev inequality implies that

supκ*<κ<κ*supz∈RN∥F(uκ)∥Lr(B(z,1))<∞ for some r>N/2.

Applying the elliptic regularity theorems to elliptic equation (4.2), by (1.3) we see that {w^κ}_{κ_*<κ<κ}^_* are uniformly bounded in R^N and equi-continuous for compact sets in R^N. Thus Proposition 6.1 follows under condition (ii). The proof of Proposition 6.1 is complete.

As a corollary of Proposition 6.1, we have:

Proposition 6.2

Assume the same conditions as in Proposition 6.1. Then there exists the limit function

(6.12) u*(x):=limκ→κ*−0uκ(x) in RN

such that u*∈BC(RN)+Lc∞(RN) and it satisfies (1.2) in R^N.

Proof

Let 0 < κ < κ′ < κ^*. Since u^κ′ and u^κ are a supersolution and the minimal solution to problem (P), respectively, we have

(6.13) uκ≤uκ′=wκ′+U1κ′≤wκ′+U1κ* in RN.

Since U1κ*∈L∞(RN) , by Proposition 6.1 we see that {u^κ} is uniformly bounded in R^N and monotone increasing with respect to κ ∈ (0, κ^*). This implies that the limit function u^* can be defined by (6.12) and it is bounded in R^N. Furthermore, we see that u^* satisfies

u*(x)=[G*F(u*)](x)+κ[G*μ](x)>0 in RN.

On the other hand, it follows that

w*(x):=u*(x)−U1κ*(x)=limκ→κ*−0[uκ(x)−U1κ(x)]=limκ→κ*−0wκ(x) in RN.

By Proposition 6.1 we apply the Arzelá–Ascoli theorem to see that w^* ∈ BC(R^N), that is, u*∈BC(RN)+Lc∞(RN) . Thus Proposition 6.2 follows.

6.2 Decay of w^*

The aim of this subsection is to prove the following proposition.

Proposition 6.3

Assume the same conditions as in Proposition 6.1. Then u^* is the minimal solution to problem (P) with κ = κ^*, that is, u^* = u^{κ^*}.

Thanks to Proposition 6.2 and (6.13), it suffices to prove that u^*(x) → 0 as |x| → ∞. In the proof, it is crucial to compare u^* with its mirror reflection with respect to hyperplanes.

For any θ ∈ S^N−1 and ℓ > 0, we define a half-space Πℓθ of R^N by

Πℓθ:={x∈RN : x⋅θ≤ℓ}.

Notice that 0∈Πℓθ and dist (0,∂Πℓθ)=ℓ . For x ∈ R^N, set

xθℓ:=x+2(ℓ−θ⋅x)θ,

which is the mirror reflection point of x with respect to the hyperplane ∂Πℓθ .

Definition 6.1

Let f be a measurable function in R^N. For R > 0, we say that f satisfies property (D_R) if

f(xℓθ)≤f(x) in Πℓθ

for θ ∈ S^N−1 and ℓ ≥ R.

The following lemma concerns with property (D_R) and the convolution of functions.

Lemma 6.3

Let f be a measurable function such that (G * f)(x) is defined for almost all R^N. If f satisfies property (D_R) for some R > 0, then G * f also satisfies property (D_R).

Proof

Assume that f satisfies property (D_R) for some R > 0. Since G is radially symmetric in R^N and monotone decreasing with respect to |x|, we have

(G*f)(x)−(G*f)(xℓθ)=∫Πℓθ(G(x−y)−G(xℓθ−y))f(y) dy−∫Πℓθ(G(x−yℓθ)−G(xℓθ−yℓθ))f(yℓθ) dy=∫Πℓθ(G(x−y)−G(xℓθ−y))(f(y)−f(yℓθ)) dy≥0

for x∈Πℓθ , θ ∈ S^N−1, and ℓ ≥ R. Here we used the relations

|x−y|=|xℓθ−yℓθ|, x,y∈RN, |x−y|≤|xℓθ−y|, x,y∈Πℓθ.

Thus Lemma 6.3 follows.

We prove Proposition 6.3 by the use of property (D_R).

Proof of Proposition 6.3

Let R > 0 be such that supp μ ⊂ B(0, R). We easily see that μ satisfies property (D_R). This together with Lemma 6.3 implies that U0κ satisfies property (D_R). Since F is monotone increasing, we see that F(U0κ) also satisfies property (D_R). Similarly, thanks to Lemma 6.3, we see that G*F(U0κ) satisfies property (D_R), which implies that U1κ satisfies property (D_R). Repeating this arguments, we deduce that Ujκ satisfies property (D_R) for j ∈ {0, 1, . . . }. Therefore, by property (W3) we see that u^κ satisfies property (D_R) for 0 < κ < κ^*, which together with (6.12) implies that u^* satisfies property (D_R). Then the following properties hold.

For any θ ∈ S^N−1, the function [R, ∞) ∋ r ↦ u^*(rθ) is monotone decreasing.
For any x ∈ R^N, there exists L > 0 such that u^*(x) ≥ u^*(y) for y ∈ R^N \ B(0, L).

Thanks to these properties, for any x₀ ∈ R^N it follows that

lim sup|x|→∞ u*(x)≤u*(x0).

Since x₀ is arbitrary, we see that

lim sup|x|→∞ u*(x)≤lim inf|x|→∞u*(x).

These imply that

(6.14) u*(x)≥u∞*:=lim|x|→∞u*(x) in RN\B(0,R).

Then it follows from Proposition 6.2 that

∫RNG(x−y)[F(u*(y))−u∞*] dy=u*(x)−u∞*−U0κ*(x)→0,

which implies that u∞*=F(u∞*) .

Assume that u∞*>0 . Then, by (F1) and (F2) we see that F′(u∞*)>1 . On the other hand, by (4.10) and (6.12) we have

(6.15) ∫RN\B(0,R)[|∇ψ|2+ψ2] dx≥∫RN\B(0,R)F′(u*)ψ2 dx

for ψ∈H01(RN\B(0,R)) . Then, by (F2) and (6.14) we obtain

∫RN\B(0,R)|∇ψ|2 dx=∫RN\B(0,R)[F′(u*)−1]ψ2 dx≥[F′(u∞*)−1]∫RN\B(0,R)ψ2 dx≥C−1∫RN\B(0,R)ψ2 dx

for ψ∈H01(RN\B(0,R)) . This means that the Poincaré inequality holds in R^N \B(0, R). This is a contradiction, and we see that u∞*=0 . Therefore, combining Proposition 6.2, we deduce that u^* is a solution to problem (P) with κ = κ^*. Thus Proposition 6.3 follows.

6.3 Proof of Theorem 1.2

We complete the proof of Theorem 1.2. It remains to prove the uniqueness of solutions to problem (P) with κ = κ^*. For this aim, we prove the following lemma.

Lemma 6.4

Assume the same conditions as in Proposition 6.1. Then λ1κ*=1 .

Proof

It follows from (F2) and (6.15) that

∫RN[|∇ψ|2+ψ2] dx≥∫RNF′(uκ*)ψ2 dx

for ψ ∈ H¹(R^N) \ {0}. This implies that λ1κ*≥1 .

Assume that λ1κ*>1 . For ∊ ∈ (0, 1] and a ∈ (0, 1], set

u¯:=U1κ*+ϵ+wκ*+aϕ1κ*∈C0(RN)+Lc∞(RN),

where wκ*:=uκ*−U1κ* . Then

(6.16) u¯−G*F(u¯)−(κ*+ϵ)G*μ=(wκ*+aϕ1κ*)−G*F(u¯)+G*F(U0κ*+ϵ)=G*[F(wκ*+U1κ*)−F(U0κ*)]−G*F(u¯)+G*F(U0κ*+ϵ)+aϕ1κ*=−G*[H(1)−H(0)]+aϕ1κ*,

where

H(x,t):=F(P(x,t))−F(Q(x,t)),P(x,t):=tU1κ*+ϵ(x)+(1−t)U1κ*(x)+wκ*(x)+atϕ1κ*(x),Q(x,t):=tU0κ*+ϵ(x)+(1−t)U0κ*(x).

Furthermore, we have

(6.17) H′(t)=F′(P(x,t))(U1κ*+ϵ−U1κ*+aϕ1κ*)−F′(Q(x,t))(U0κ*+ϵ−U0κ*)=F′(uκ*)aϕ1κ*+{F′(P(x,t))−F′(uκ*)}aϕ1κ*+F′(P(x,t))(V1κ*+ϵ−V1κ*)+{F′(P(x,t))−F′(Q(x,t))}(U0κ*+ϵ−U0κ*)

for x ∈ R^N and t ∈ (0, 1).

On the other hand, it follows from property (W2) that 0 < w^{κ^*} (x) ≤ Cg(x) in R^N. Then, by (3.3) and (4.8) we have

(6.18) 0≤P(x,t)−Q(x,t)=tV1κ*+ϵ(x)+(1−t)V1κ*(x)+wκ*(x)+atϕ1κ*(x) ≤V1κ*+1(x)+wκ*(x)+ϕ1κ*(x)≤Cg(x),max{P(x,t),Q(x,t)}≤U1κ*+ϵ(x)+wκ*(x)+atϕ1κ*(x) ≤CU1κ*(x)+Cg(x)=C[U0κ*(x)+V1κ*(x)]+Cg(x)≤Cg(x),min{P(x,t),Q(x,t)}≥C−1g(x).

These together with (3.2) and (3.3) imply that

(6.19) 0≤{F′(P(x,t))−F′(Q(x,t))}(U0κ*+ϵ(x)−U0κ*(x))≤Cϵg(x)α+1

for x ∈ R^N and t ∈ (0, 1). Furthermore, by Lemma 3.1 we have

(6.20) F′(P(x,t))(V1κ*+ϵ−V1κ*)≤Cϵαg(x)α+1

for x ∈ R^N and t ∈ (0, 1). Similarly to (6.18), we have

0≤P(x,t)−[U1κ*(x)+wκ*(x)]=t[U1κ*+ϵ(x)−U1κ*(x)]+atϕ1κ*(x) ≤CϵU1κ*(x)+Cag(x)≤C(ϵ+a)g(x),max{P(x,t),U1κ*(x)+wκ*(x)}≤U1κ*+ϵ(x)+wκ*(x)+aϕ1κ*(x)≤Cg(x),min{P(x,t),U1κ*(x)+wκ*(x)}≥C−1g(x).

These imply that

(6.21) 0≤F′(P(x,t))−F′(uκ*)≤C(ϵ+a)g(x)α

for x ∈ R^N and t ∈ (0, 1). Combining (6.17), (6.19), (6.20), and (6.21), we obtain

H′(x,t)≤aF′(uκ*(x))ϕ1κ*(x)+Ca(ϵ+a)g(x)α+1+C(ϵα+ϵ)g(x)α+1

for x ∈ R^N and t ∈ (0, 1). Therefore we deduce from (2.3), (4.8), (4.9), and (6.16) that

u¯(x)−[G*F(u¯)](x)−(κ*+ϵ)[G*μ](x)≥−a[G*(F′(uκ*)ϕ1κ*)](x)−C[a(ϵ+a)+ϵα+ϵ][G*gα+1](x)+aϕ1κ*(x)≥a(1−1λ1κ*)ϕ1κ*(x)−C[a(ϵ+a)+ϵα+ϵ]g(x)≥a(1−1λ1κ*)ϕ1κ*(x)−C[a(ϵ+a)+ϵα+ϵ]ϕ1κ*(x)

in R^N. Then, taking suitable small enough a > 0 and ∊ > 0, we observe that

u¯(x)−[G*F(u¯)](x)−(κ*+ϵ)[G*μ](x)>0 in RN.

This means that ū is a supersolution to problem (P) with κ = κ^* + ∊. Then, by property (W3) we find a solution to problem (P) with κ = κ^* + ∊. This contradicts the definition of κ^*. Therefore we deduce that λ1κ*=1 , and the proof is complete.

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2

Thanks to Proposition 6.3, it suffices to prove the uniqueness of solutions to problem (P) with κ = κ^*.

Set wκ*=uκ*−U1κ* . Let ũ be a solution to problem (P) with κ = κ^* such that ũ ≠ u^{κ^*}. Since u^{κ^*} is the minimal solution, we have ũ ≥ u^{κ^*} and ũ ≢ u^{κ^*} in R^N. Then it follows from (1.2) and the elliptic regularity theorems (see e.g. [31, Theorems 1 and 7]) that ũ − u^{κ^*} ∈ C(R^N) and

(6.22) u˜(x)−uκ*(x)>0 in RN.

Set w˜=u˜−U1κ* . Then w^{κ^*} and w˜ are solutions to (4.2) with κ = κ^* and the function z:=w˜−wκ* satisfies

(6.23) z>0 in RN, −Δz+z=F(w˜+U1κ*)−F(wκ*+U1κ*) in RN.

Let ϕ1κ* be the eigenfunction to problem (E_κ^_*) corresponding to λ1κ* such that ϕ1κ*>0 in R^N. By Lemma 6.4 we have

(6.24) −Δϕ1κ*+ϕ1κ*=F′(wκ*+U1κ*)ϕ1κ* in RN.

Multiplying (6.24) by z and integrating it on R^N, we obtain

∫RN[∇z∇ϕ1κ*+zϕ1κ*] dx=∫RNF′(wκ*+Uj*κ*)ϕ1κ*(w˜−wκ*) dx.

On the other hand, by (6.23) we see that

∫RN[∇z∇ϕ1κ*+zϕ1κ*] dx=∫RN[F(w˜+U1κ*)−F(wκ*+U1κ*)]ϕ1κ* dx.

These imply that

∫RN[F(w˜+U1κ*)−F(wκ*+U1κ*)−F′(wκ*+U1κ*)(w˜−wκ*)]ϕ1κ* dx=0,

that is,

(6.25) ∫RN[F(u˜)−F(uκ*)−F′(uκ*)(u˜−uκ*)]ϕ1κ* dx=0.

On the other hand, it follows from condition (F2) that F(t) ≥ F(s) + F′(s)(t − s) for t ≥ s. Then, by (6.22) and (6.25) we see that

(6.26) F(u˜)−F(uκ*)−F′(uκ*)(u˜−uκ*)=0 in RN.

Consider the case when t_* > 0, where t_* is as in Lemma 4.1. By assumption (i) of Theorem 1.2 with (3.3) and Lemma 6.2 we have G * μ ∈ C₀(R^N). Recalling that uκ*=U1κ*+wκ* , we see that u^{κ^*} ∈ C₀(R^N). On the other hand, it follows from Lemma 4.1 (i) that κ^* > κ_*, which implies that

(6.27) ∥uκ*∥L∞(RN)≥∥U1κ*∥L∞(RN)≥∥U0κ*∥L∞(RN)=κ*∥G*μ∥L∞(RN)>κ*∥G*μ∥L∞(RN)=t*.

Then we find x_* ∈ R^N such that u^{κ^*} (x_*) = t_*, which implies that

F(u˜(x*))=F(u˜(x*))−F(uκ*(x*))−F′(uκ*(x*))(u˜(x*)−uκ*(x*))=0.

Then we have ũ(x_*) ≤ t_*, which contradicts (6.22). Thus u^{κ^*} is the unique solution to problem (P) with κ = κ^* in the case when t_* > 0.

Consider the case when t_* = 0. Since F(0) = 0 and F(t) > 0 if t > 0, by the convexity of F we find a sequence {t_j} ⊂ (0, ∞) such that lim_j→∞ t_j = 0 and F″(t_j) > 0. This together with u^{κ^*} ∈ C₀(R^N) implies that

(6.28) F″(uκ*(x′))>0 for some x′∈ RN .

Then, by (F2), (6.22), and (6.28) we have

F(u˜(x′))−F(uκ*(x′))−F′(uκ*(x′))(u˜(x′)−uκ*(x′))=∫01(1−τ)d2dτ2F(τu˜(x′)+(1−τ)uκ*(x′)) dτ=(u˜(x′)−uκ*(x′))2∫01(1−τ)F″(τu˜(x′)+(1−τ)uκ*(x′)) dτ>0.

This contradicts (6.26), and we see that u^{κ^*} is the unique solution to problem (P) with κ = κ^* in the case when t_* = 0. Thus Theorem 1.2 follows.

At the end of this paper we give other sufficient conditions for the uniqueness of solutions to problem (P) with κ = κ^* in the case when lim sup_t→∞ F′(t)/F(t) < ∞. We remark that G * μ is not necessarily continuous in R^N under the assumptions of Theorem 6.1.

Theorem 6.1

Assume conditions (F1) and (F2). Let μ∈Lc1(RN)\{0} be nonnegative in R^N, and assume (1.3). Let 2 ≤ N ≤ 9 and lim sup_t→∞ F′(t)/F(t) < ∞. Then problem (P) with κ = κ^* possesses a unique solution if either t_* = 0 or F is strictly convex in (t_*, ∞).

Proof

Similarly to Theorem 1.2, it suffices to prove the uniqueness of solutions to problem (P) with κ = κ^*. Consider the case when t_* = 0. Since μ has a compact support, we see that G * μ is continuous in R^N \ B(0, R) for some R > 0. Then we apply the same argument as in the proof of Theorem 1.2 in the case when t_* = 0 to see that identity (6.26) does not hold. On the other hand, if F is strictly convex in (t_*, ∞), then, by (6.27) we see that identity (6.26) does not hold. Then, similarly to the proof of Theorem 1.2, we complete the proof.

Acknowledgements

The first and the second authors were supported in part by JSPS KAKENHI Grant Number JP19H05599. The second author was supported in part by JSPS KAKENHI Grant Number 20KK0057 and 21H00990.

Conflict of interest statement:
The authors state no conflict of interest.

References

[1] N. Ba, Y.-y. Wang, and L. Zheng, Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent, Acta Math. Appl. Sin. Engl. Ser. 32 (2016), 81–94.10.1007/s10255-015-0537-6Search in Google Scholar

[2] M. Basiri and A. Moameni, Solutions of supercritical semilinear non-homogeneous elliptic problems, Nonlinear Anal. 165 (2017), 121–142.10.1016/j.na.2017.09.014Search in Google Scholar

[3] D.-M. Cao and H.-S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in R^N, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 443–463.10.1017/S0308210500022836Search in Google Scholar

[4] K.-J. Chen, Exactly two entire positive solutions for a class of nonhomogeneous elliptic equations, Differential Integral Equations 17 (2004), 1–16.10.57262/die/1356060469Search in Google Scholar

[5] K.-J. Chen and C.-C. Peng, Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems, J. Differential Equations 240 (2007), 58–91.10.1016/j.jde.2007.05.023Search in Google Scholar

[6] D. G. de Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal. 8 (1984), no. 6, 655–665.10.21236/ADA130574Search in Google Scholar

[7] D. G. de Figueiredo, J. M. do Ó, and B. Ruf, Elliptic equations and systems with critical Trudinger-Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), 455–476.10.3934/dcds.2011.30.455Search in Google Scholar

[8] L. R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth, Nonlinear Anal. 95 (2014), 607–624.10.1016/j.na.2013.10.010Search in Google Scholar

[9] M. del Pino, Nonlinear elliptic problems above criticality, Milan J. Math. 74 (2006), 313–338.10.1007/s00032-006-0058-0Search in Google Scholar

[10] M. del Pino, Supercritical elliptic problems from a perturbation viewpoint, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 69–89.10.3934/dcds.2008.21.69Search in Google Scholar

[11] Y. Deng and Y. Li, Existence and bifurcation of the positive solutions for a semilinear equation with critical exponent, J. Differential Equations 130 (1996), 179–200.10.1006/jdeq.1996.0138Search in Google Scholar

[12] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations 2 (1997), 361–382.10.57262/ade/1366742248Search in Google Scholar

[13] Y. Deng, Y. Ma, and C. X. Zhao, Existence and properties of multiple positive solutions for semi-linear equations with critical exponents, Rocky Mountain J. Math. 35 (2005), 1479–1512.10.1216/rmjm/1181069647Search in Google Scholar

[14] Y. Deng, S. Peng, and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent, Discrete Contin. Dyn. Syst. 32 (2012), 795–826.10.3934/dcds.2012.32.795Search in Google Scholar

[15] J. M. do Ó, E. Medeiros, and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), no. 1, 286–304. MR242265110.1016/j.jmaa.2008.03.074Search in Google Scholar

[16] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of ℝ^N, J. Math. Pures Appl. (9) 87 (2007), 537–561.10.1016/j.matpur.2007.03.001Search in Google Scholar

[17] C. Gui, W.-M. Ni, and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rⁿ, Comm. Pure Appl. Math. 45 (1992), 1153–1181.10.1002/cpa.3160450906Search in Google Scholar

[18] C. Gui, W.-M. Ni, and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations 169 (2001), 588–613.10.1006/jdeq.2000.3909Search in Google Scholar

[19] N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal. 29 (1997), 889–901.10.1016/S0362-546X(96)00176-9Search in Google Scholar

[20] N. Hirano and W. S. Kim, Multiple existence of solutions for a nonhomogeneous elliptic problem with critical exponent on ℝ^N, J. Differential Equations 249 (2010), 1799–1816.10.1016/j.jde.2010.05.003Search in Google Scholar

[21] K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. (9) 128 (2019), 183–212.10.1016/j.matpur.2019.04.003Search in Google Scholar

[22] L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differential Integral Equations 10 (1997), 609–624.10.57262/die/1367438634Search in Google Scholar

[23] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269.10.1007/BF00250508Search in Google Scholar

[24] Y. Miyamoto, Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball, Ann. Mat. Pura Appl. (4) 194 (2015), 931–952.10.1007/s10231-014-0404-8Search in Google Scholar

[25] Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth, J. Differential Equations 264 (2018), 2684–2707.10.1016/j.jde.2017.10.034Search in Google Scholar

[26] Y. Miyamoto, Infinitely many non-radial singular solutions of Δu + e^u = 0 in ℝ^N \ {0}, 4 ⩽ N ⩽ 10, Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), 133–147.10.1017/S0308210517000051Search in Google Scholar

[27] Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms, J. Differential Equations 235 (2007), 439–483.10.1016/j.jde.2007.01.006Search in Google Scholar

[28] Y. Naito and T. Sato, Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent, Ann. Mat. Pura Appl. (4) 191 (2012), 25–51.10.1007/s10231-010-0173-ySearch in Google Scholar

[29] P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745–771.10.1007/s00208-003-0469-ySearch in Google Scholar

[30] T. Sato and T. Watanabe, Singular positive solutions for a fourth order elliptic problem in ℝ^N, Commun. Pure Appl. Anal. 10 (2011), 245–268.10.3934/cpaa.2011.10.245Search in Google Scholar

[31] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.10.1007/BF02391014Search in Google Scholar

[32] Q. Wang, Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in ℝ^N, Commun. Pure Appl. Anal. 15 (2016), 1671–1688.10.3934/cpaa.2016008Search in Google Scholar

[33] X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Anal. 12 (1988), 1297–1316.10.1016/0362-546X(88)90061-2Search in Google Scholar

[34] X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations 92 (1991), 163–178.10.1016/0022-0396(91)90045-BSearch in Google Scholar

[35] X. P. Zhu and H. S. Zhou, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 301–318.10.1017/S0308210500020667Search in Google Scholar

Received: 2021-09-13

Accepted: 2021-12-13

Published Online: 2022-02-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity

Abstract

1 Introduction

Definition 1.1

Theorem 1.1

Theorem 1.2

2 Preliminaries

2.1 Fundamental solution G

2.2 Eigenvalue problem

Lemma 2.1

3 Approximate solutions

Lemma 3.1

Proof

4 Proof of Theorem 1.1

Lemma 4.1

Proof

Lemma 4.2

Proof

Proof of Theorem 1.1

5 Some inequalities related to (F1) and (F2)

Lemma 5.1

Proof

Lemma 5.2

Proof

Lemma 5.3

Proof

Lemma 5.4

Proof

Lemma 5.5

Proof

6 Proof of Theorem 1.2

6.1 Uniform estimates of wκ

Proposition 6.1

Lemma 6.1

Proof

Proof of Proposition 6.1 under condition (i)

Lemma 6.2

Proof

Proof of Proposition 6.1 under condition (ii)

Proposition 6.2

Proof

6.2 Decay of w*

Proposition 6.3

Definition 6.1

Lemma 6.3

Proof

Proof of Proposition 6.3

6.3 Proof of Theorem 1.2

Lemma 6.4

Proof

Proof of Theorem 1.2

Theorem 6.1

Proof

Acknowledgements

References

Journal and Issue

Articles in the same Issue

6.1 Uniform estimates of w^κ

6.2 Decay of w^*