Startseite Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
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Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

  • Divya Goel und Konijeti Sreenadh EMAIL logo
Veröffentlicht/Copyright: 6. August 2019
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 9 Heft 1

Abstract

The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation

−Δu=∫Ω|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u+finΩ,u=0 on ∂Ω,

where Ω is a smooth bounded annular domain in ℝN(N ≥ 3), 2μ∗=2N−μN−2 , f ∈ L∞(Ω) and f ≥ 0. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of the annulus is sufficiently small.

Keywords: Hardy-Littlewood-Sobolev inequality; critical problems; non-contractible domains
MSC 2010: 35A15; 35J60; 35J20

1 Introduction

In the pioneering work, Tarantello [31] studied the nonhomogeneous elliptic equation

−Δu=|u|2∗−2u+f in Ω,u=0 on ∂Ω, (1.1)

where 2∗=2NN−2 is the critical Sobolev exponent and Ω is a bounded domain in ℝN with smooth boundary. If f ∈ H−1 then it is shown that there exists at least two solutions of (1.1) by using variational methods. Cao and Zhou [9] proved the existence of two positive solutions of the following nonhomogeneous elliptic equation

−Δu=f(x,u(x))+h in RN (1.2)

where f(x, u) is a Carathéodory function with subcritical grotwh at ∞. Further, many researchers investigated (1.1) and (1.2) for the existence and multiplicity of solutions. For details, we refer [10, 11, 20, 21, 33] and references therein. Recently, Gao and Yang [30] proved the existence of two positive solutions of the nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent using the splitting Nehari manifold method of Tarantello [31].

The existence, uniqueness, and multiplicity of positive solutions of the nonlocal elliptic equation, precisely the Choquard equation both for mathematical analysis and in perspective of physical models has recently gained significant attention amongst researchers. As an instance, in 1954 Pekar [28] proposed the equation

−Δu+u=1|x|∗|u|2u in R3 (1.3)

to study the quantum theory of polaron. Later in 1976, Ph. Choquard [22] examined the steady state of one component plasma approximation in Hartee-Fock theory using (1.3). In [22], Leib proved the existence and uniqueness of the ground state of (1.3). The work of Moroz and Schaftingen enriches the literature of Choquard equations. In [25] authors studied the following Choquard equation

−Δu+Vu=Iα∗F(u)F′(u),in RN, (1.4)

where α ∈ (0, N), N ≥ 3, Iα is the Riesz Potential and F(u) ∈ C1(ℝ, ℝ) with sub critical growth. In this work authors established the existence of ground state soloutions of (1.4) and assuming some suitable growth conditions on F and V, they studied the properties like constant sign solutions and radial symmetry of the solution. Moreover, authors proved the Pohožaev identity and nonlocal Brezis-Kato type estimate. Interested readers are referred to [16, 24, 26, 27] and references therein for the study of Choquard equation on the unbounded domain.

Concerning the boundary value problems of Choquard equation, Gao and Yang [15] studied the Brezis-Nirenberg type existence results for the following critical equation

−Δu=λh(u)+∫Ω|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u in Ω,u=0 on ∂Ω,

where λ > 0, 0 < μ < N, h(u) = u, Ω is a smooth bounded domain in ℝN. Later in [14] authors proved the existence and multiplicity of positive solutions for convex and convex-concave type nonlinearities (h(u) = uq, 0 < q < 1) using variational methods.

The geometry of the domain Ω plays an essential and significant role on the existence and multiplicity of the elliptic boundary value problems. Indeed, in [12], Coron proved the existence of a high energy positive solution of the problem

−Δu=|u|2∗−2uinΩ,u=0 on ∂Ω, (1.5)

where Ω is a bounded domain in ℝN(N ≥ 3), precisely an annulus with a small hole. Later in [3], Bahri and Coron, proved that a positive solution always exists as long as the domain has non-trivial homology with ℤ2-coefficients. In [6], Benci and Cerami studied the following equation

−εΔu+u=f(u)inΩ,u=0 on ∂Ω, (1.6)

where ε ∈ ℝ+, Ω is a bounded domain in ℝN(N ≥ 3) and f : ℝ+ → ℝ is a C1,1 function. Here authors proved that there exists ϵ* > 0 such that for all ε ∈ (0, ϵ*), (1.6) has cat(Ω)+1 solutions under some growth conditions on the function f. Since then, the study of existence and multiplicity of solutions of elliptic equations over non-contractible domain has been substantially studied, for instance, [4, 5, 13, 20, 29, 32] and references therein.

The existence of high energy solution of (1.5) is a much more delicate issue. In this spirit, recently Goel, Rădulescu and Sreenadh [19] studied the Coron problem for Choquard equations. Here authors proved the existence of a positive high energy solution for the problem (Pf) when f(x) ≡ 0 and Ω is a smooth bounded domain in ℝN(N ≥ 3) satisfying the following condition

  1. There exists constants 0 < R1 < R2 < ∞ such that

    {x∈RN:R1<|x|<R2}⊂Ω,{x∈RN:|x|<R1}⊈Ω¯.

In the light of above works, in this article, we study following problem

(Pf)−Δu=∫Ω|u+(y)|2μ∗|x−y|μdy|u+|2μ∗−2u++f,in Ω,u=0 on ∂Ω,

where 2μ∗=2N−μN−2 , is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality (2.1) and f ∈ F̂ with F̂ := {f : f ∈ L∞(Ω), f ≥ 0, f ≢ 0}. The domain Ω ⊂ ℝN(N ≥ 3) satisfies the condition (A). Here we prove the existence of four solutions of the problem (Pf). To achieve this, we first seek the help of Nehari manifold associated with (Pf) to prove the existence of the first solution (say u1). To proceed further, we prove many new estimates on the convolution terms involving the minimizers of best constant SH,L (see Lemma 4.1, 4.3 and 4.4). With the help of these estimates we prove that the minima of the functional over 𝓝f is below the first critical level where the first critical level is

Jf(u1)+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Here 𝓙f is the energy functional associated to (Pf) (defined in (2.3)). Moreover, 𝓙f satisfies the Palais-Smale condition below the first critical level. Subsequently, we show the existence of the second and the third solution of (Pf), in Nf− (a closed subset of the Nehari manifold) by using a well-known result of Ambrosetti [2](see Lemma 5.2) and assumption (A). To study the existence of the fourth solution, a high energy solution, we prove that the functional 𝓙f satisfies the Palais-Smale condition between the first and the second critical levels, where the second critical level is

infu∈Nf−Jf(u)+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

To prove the existence of fourth solution, we use the minmax Lemma (See Lemma 6.6). To the best of our knowledge, there is no work on the existence and multiplicity of solutions to Choquard equations (Pf) in non-contractible domains. With this introduction, we state our main result.

Theorem 1.1

Assume μ < min{4, N}, f ∈ L∞(Ω) and f ≥ 0 and Ω be a bounded domain satisfying the conditon (A). Then there exists e* > 0 such that (Pf) has at least three positive solutions whenever 0 < ∥f∥H−1 < e*. Moreover, if R1 is small enough then there exists e** > 0 such that (Pf) has at least four positive solutions whenever 0 < ∥f∥H−1 < e**.

The paper is organized as follows: In Section 2, we give the variational framework and preliminary results. In section 3, using the Nehari manifold technique, we prove the existence of the first solution. In section 4, we prove some crucial estimates of the minimizer of SH,L(defined in (2.2)) and analyze the Palais-Smale sequences. In section 5, we prove the existence of the second and third solution. In section 6, we prove the existence of the fourth solution.

2 Variational framework and preliminary results

We start with the familiar Hardy-Littlewood-Sobolev Inequality which leads to the study of nonlocal Choquard equation using variational methods.

Proposition 2.1

[23](Hardy-Littlewood-Sobolev Inequality) Let t, r > 1 and 0 < μ < N with 1/t + μ/N + 1/r = 2, f ∈ Lt(ℝN) and h ∈ Lr(ℝN). There exists a sharp constant C(t, r, μ, N) independent of f and h such that

∫RN∫RNf(x)h(y)|x−y|μ dxdy≤C(t,r,μ,N)∥f∥Lt(RN)∥h∥Lr(RN). (2.1)

If t = r = 2N/(2N − μ), then

C(t,r,μ,N)=C(N,μ)=πμ2Γ(N2−μ2)Γ(N−μ2)Γ(N2)Γ(μ2)−1+μN.

Equality holds in (2.1) if and only if f ≡ (constant)h and

h(x)=A(y2+|x−a|2)(2N−μ)/2,

for some A ∈ ℂ, 0 ≠ y ∈ ℝ and a ∈ ℝN.□

The best constant for the embedding D1,2(ℝN) into L2*(ℝN) (where 2∗=2NN−2 )is defined as

S=infu∈D1,2(RN)∖{0}∫RN|∇u|2dx:∫RN|u|2∗dx=1.

Consequently, we define

SH,L=infu∈D1,2(RN)∖{0}∫RN|∇u|2dx:∫RN∫RN|u(x)|2μ∗|u(y)|2μ∗|x−y|μ dxdy=1. (2.2)

Lemma 2.2

[15] The constant SH,L defined in (2.2) is achieved if and only if

u=Cbb2+|x−a|2N−22

where C > 0 is a fixed constant, a ∈ ℝN and b ∈ (0, ∞) are parameters. Moreover,

S=SH,L (C(N,μ))N−22N−μ.

Lemma 2.3

[15] For N ≥ 3 and 0 < μ < N. Then

∥.∥NL:=∫RN∫RN|.|2μ∗|.|2μ∗|x−y|μ dxdy12.2μ∗

defines a norm on L2*(ℝN).

The energy functional 𝓙f : H01 (Ω) → ℝ associated with the problem (Pf) is

Jf(u)=12∫Ω|∇u|2 dx−12.2μ∗∫Ω∫Ω|u+(x)|2μ∗|u+(y)|2μ∗|x−y|μ dxdy−∫Ωfu dx, (2.3)

where u+ = max(u, 0). By using Hardy-Littlewood-Sobolev inequality (2.1), we have

∫Ω∫Ω|u+(x)|2μ∗|u+(y)|2μ∗|x−y|μ dxdy12μ∗≤C(N,μ)2N−μN−2|u|2∗2.

It is not difficult to show that the functional 𝓙f ∈ C1( H01 (Ω), ℝ) and moreover, if μ < min {4, N} then 𝓙f ∈ C2( H01 (Ω), ℝ).

Definition 2.4

A function u ∈ H01 (Ω) is called a weak solution of the problem (Pf) if for all v ∈ H01 (Ω) the following holds

∫Ω∇u⋅∇v dx−∫Ω∫Ω|u+(x)|2μ∗|u+(y)|2μ∗−1v(y)|x−y|μ dxdy−∫Ωfv dx=0.

Definition 2.5

For c ∈ ℝ, {un} is a (PS)c sequence in H01 (Ω) for 𝓙f if 𝓙f = c + o(1) and Jf′ (un) = o(1) strongly in H−1 as n → ∞. We say 𝓙f satisfies the (PS)c condition in H01 (Ω) if every (PS)c sequence in H01 (Ω) has a convergent subsequence.

Since 𝓙f is not bounded below on H01 (Ω), it is worth to consider the Nehari manifold

Nf:={u∈H01(Ω)∖{0}|u+≢0 and 〈Jf′(u),u〉=0},

where 〈, 〉 denotes the usual duality. We define

Υf=infu∈NfJf(u).

Note that when f(x) ≡ 0, Υ0(Ω) is independednt of Ω and Υ0(Ω):=Υ0=N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Notations: Throughout the paper we will use the notation 𝓙0 = 𝓙, 𝓝0 = 𝓝, ∥.∥ = ∥.∥H01(Ω)

a(u)=∫Ω|∇u|2 dxandb(u)=∫Ω∫Ω(u+(x))2μ∗(u+(y))2μ∗|x−y|μ dxdy.

An easy consequence of (2.1) gives 𝓙f is coercive and bounded below on 𝓝f.

Proposition 2.6

For any u, v ∈ H01 (Ω), we have

∫Ω∫Ω|u(x)|2μ∗|v(y)|2μ∗|x−y|μ dxdy≤∫Ω∫Ω|u(x)|2μ∗|u(y)|2μ∗|x−y|μ dxdy12∫Ω∫Ω|v(x)|2μ∗|v(y)|2μ∗|x−y|μ dxdy12.

Proof

For details of the proof, see [17, Lemma 2.3].□

Lemma 2.7

For each u ∈ H01 (Ω), there exists a unique t > 0 such that t u ∈ 𝓝. Moreover, there holds Υ0≤N−μ+22(2N−μ)a(u)2μ∗b(u)12μ∗−1.

Proof

Let mu(t)=t22a(u)−t2.2μ∗2.2μ∗b(u) then on solving mu′(t) = 0, we get unique t(u)=a(u)b(u)12(2μ∗−1) such that t(u)u ∈ 𝓝. From the definition of Υ0, we have

Υ0≤J(t(u)u)=12−12.2μ∗a(u)b(u)12μ∗−1a(u)=N−μ+22(2N−μ)a(u)2μ∗b(u)12μ∗−1.

□

Remark 2.8

We remark that by [15, Lemma 1.3], SH,L is never achieved on bounded domain. Therefore if u is a solution of the following equation

−Δu=∫Ω|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u in Ω,u=0 on  ∂Ω,

then J(u)>Υ0=N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Lemma 2.9

A sequence {un} is a (PS)Υ0- sequence for 𝓙 in H01 (Ω) if and only if 𝓙(un) = Υ0 + on(1) and a(un) = b(un) + on(1).

Proof

Clearly, any (PS)Υ0- sequence satisfies a(un) = b(un) + on(1) and 𝓙(un) = Υ0 + on(1). Conversely, let 𝓙(un) = Υ0 + on(1) and a(un) = b(un) + on(1) then Υ0 = 𝓙(un) = N−μ+22(2N−μ)b(un) + on(1) and hence we have

b(un)=DΥ0+on(1) where D=2(2N−μ)N−μ+2. (2.4)

Define Tn(ψ)=∫Ω∫Ω(un+(x))2μ∗(un+(y))2μ∗−1ψ(y)|x−y|μ dxdy for ψ ∈ H01 (Ω) and n = 1, 2, ….

Claim: ∥Tn∥H−1=(DΥ0)12+on(1).

Let ψ ∈ H01 (Ω) such that ∥ψ∥ = 1 then by Lemma 2.7, we know that there exists a t > 0 such that a(tψ) = b(tψ). Therefore, t=∥ψ∥NL−2μ∗2μ∗−1  and  Υ0≤1D∥ψ∥NL−2.2μ∗2μ∗−1. This implies,

∥ψ∥NL≤1DΥ02μ∗−12.2μ∗. (2.5)

Taking into account (2.4), (2.5), Proposition 2.6 and employing Hölder’s inequality, for each n, we have

|Tn(ψ)|≤∫Ω∫Ω(un+(x))2μ∗(un+(y))2μ∗|x−y|μ dxdy2.2μ∗−12.2μ∗∫Ω∫Ω|ψ(x)|2μ∗|ψ(y)|2μ∗|x−y|μ dxdy12.2μ∗=b(un)2.2μ∗−12.2μ∗∥ψ∥NL≤1DΥ02μ∗−12.2μ∗(DΥ0+on(1))2.2μ∗−12.2μ∗=(DΥ0)12+on(1) as n→∞.

So, we get ∥Tn∥H−1≤(DΥ0)12+on(1). Moreover, Tnun∥un∥=(b(un))12=(DΥ0)12+on(1). This implies ∥Tn∥H−1 = (DΥ0)12+on(1). Hence the proof of claim follows. Now, by Riesz representation theorem, for each n, there exists vn ∈ H01 (Ω) such that

Tn(ψ)=〈vn,ψ〉=∫Ω∇vn⋅∇ψ dx and ∥vn∥=∥Tn∥H−1=(DΥ0)12+on(1).

Thus, 〈vn, un〉 = Tn(un) = b(un) = DΥ0 + on(1). Hence,

∥un−vn∥2=∥un∥2−2〈un,vn〉+∥vn∥2=DΥ0−2DΥ0+DΥ0+on(1)=on(1) as n→∞.

For any ψ ∈ H01 (Ω) with ∥ψ∥ = 1, we have

〈J′(un),ψ〉=∫Ω∇un⋅∇ψ dx−Tn(ψ)=〈un,ψ〉−〈vn,ψ〉=〈un−vn,ψ〉.

Therefore, ∥𝓙′(un)∥H−1 ≤ ∥un − vn∥ = on(1). It implies 𝓙′(un) → 0 in H−1.□

Clearly, 𝓝f contains every non zero solution of (Pf) and we know that the Nehari manifold is closely related to the behavior of the fibering maps ϕu : ℝ+ → ℝ defined as ϕu(t) = 𝓙f(tu). It is easy to see that tu ∈ 𝓝f if and only if ϕu′ (t) = 0 and elements of 𝓝f correspond to stationary points of the fibering maps. It is natural to divide 𝓝f into the following sets

Nf+=:{u∈Nf|ϕu′′(1)>0},Nf−=:{u∈Nf|ϕu′′(1)<0},andNf0=:{u∈Nf|ϕu′′(1)=0}.

We also denote the infimum over Nf+  and  Nf− as

Υf+=infu∈Nf+Jf(u)Υf−=infu∈Nf−Jf(u).

3 Existence of First Solution

In this section we prove the existence of first solution by showing the existence of minimizer for 𝓙f over the Nehari manifold 𝓝f. First we state some Lemmas whose proof can be found in [30]. We further prove some properties of the manifold Nf+ .

Lemma 3.1

If f ∈ F̂ and ∥f∥H−1 < e00 := CN,μSH,L2μ∗2.2μ∗−2 where CN,μ=12.2μ∗−12.2μ∗−12.2μ∗−2(2.2μ∗−2) then α0:=infu∈E{CN,μ∥u∥2.2μ∗−12μ∗−1−∫Ωfu dx} is acheived, where

E:={u∈H01(Ω):∫Ω∫Ω|u(x)|2μ∗|u(y)|2μ∗|x−y|μ dxdy=1}.

Proof

Proof follows from [30, Lemma 4.1]. Since we consider λ = 0 in equation (4.1) of [30], our result holds for all N ≥ 3.□

Lemma 3.2

For every u ∈ 𝓝f, u ≢ 0 we have a(u)−(2.2μ∗−1)b(u)≠0. In particular, Nf0 = {0}.

Lemma 3.3

For each u ∈ H01 (Ω) with u+ ≢ 0 the following holds:

  1. There exists a unique t− = t−(u) > 0 such that t− u Nf− (Ω). In particular,

    t−>a(u)(2.2μ∗−1)b(u)12.2μ∗−2:=tmax

    and Jf(t−u)=maxt≥tmaxJf(tu).

  2. If ∫Ωfu>0 , then there exists unique t+ ∈ (0, tmax) such that t+ u ∈ Nf+ (Ω) and

    Jf(t+u)=min0<t≤t−Jf(tu).

  3. t−(u) is a continuous function.

  4. Nf−={u∈H01(Ω)∖{0}|u+≢0  and  1∥u∥t−(u∥u∥)=1}.

Lemma 3.4

For each u ∈ Nf+ (Ω), we have ∫Ω fu dx > 0 and 𝓙f(u) < 0. In particular, Υf(Ω)≤Υf+(Ω)<0.

Lemma 3.5

Let u ∈ 𝓝f(Ω) be such that Jf(u)=minw∈Nf(Ω)Jf(w) = Υf(Ω) then ∫Ω fu dx > 0 and u is a solution of (Pf).

Lemma 3.6

𝓙f has Palais-Smale sequences at each of the levels Υf(Ω), Υf+(Ω)  and  Υf−(Ω).

Lemma 3.7

Let {un} ∈ 𝓝f be a (PS)Υf(Ω) sequence for 𝓙f, then there exists a subsequence of {un}, still denoted by {un}, and a non-zero u1 ∈ H01 (Ω) such that un → u1 strongly in H01 (Ω). Moreover, u1 ∈ 𝓝f and is a solution to (Pf).

Proof

𝓙f is bounded below and coercive implies {un} is bounded in H01 (Ω). So, there exists a subsequence still denoted by {un} such that un ⇀ u1 weakly in H01 (Ω). By [19, Lemma 4.2], we have Jf′ (u1) = 0. In particular, u1 ∈ 𝓝f and Jf(u1)=12−12.2∗μa(u1)−1−12.2∗μ∫Ωfu1 dx. Now, using the fact that a is weakly lower semi continuous we have

Υf(Ω)≤Jf(u1)≤lim infn→∞12−12.2∗μa(un)−limn→∞1−12.2∗μ∫Ωfun dx=Υf(Ω).

Consequently, we have Υf(Ω) = 𝓙f(u1). Let wn = un − u1 then by [19, Lemma 4.1], [15, Lemma 2.2] and the fact that Jf′ (u1) = 0, we obtain 𝓙f(wn) = 𝓙f(un) − 𝓙f(u1) = on(1) and 〈Jf′(wn),ϕ〉=〈Jf′(un),ϕ〉−〈Jf′(u1),ϕ〉+on(1) = on(1). Therefore, 〈 Jf′ (wn), wn〉 = on(1). It implies Jf(wn)=12−12.2∗μa(wn)−∫Ωfwn dx=on(1) and since ∫Ω fwn dx = on(1), we get a(wn) = on(1). Hence un → u strongly in H01 (Ω).□

Lemma 3.8

If u be a solution of (Pf) then u ∈ C2(Ω). Moreover, u is a positive solution.

Proof

Let u be a solution of (Pf) and G(x, u) = ∫Ω|u+(y)|2μ∗|x−y|μdy|u+|2μ∗−2u+f . By using same assertions and arguments as in [25, Proposition 3.1 and Theorem 2], we have ∫Ω|u+(y)|2μ∗|x−y|μdy ∈ L∞ (Ω) and since f ∈ F̂, we have |G(x, u)| ≤ C(1 + |u|2*−1). Then by the standard elliptic regularity u ∈ C2(Ω). Since f ≥ 0, we get u ≥ 0 and by using strong maximum principle, u is a positive solution of (Pf).□

Lemma 3.9

Let μ < min{ 4, N} and k0=12.2μ∗−112(2μ∗−1)SH,L2μ∗2(2μ∗−1) and f ∈ F̂, ∥f∥H−1 ≤ e00 (where e00 is defined in Lemma 3.1) then

  1. Nf+ (Ω) ⊂ Bk0(0) := {u ∈ H01 (Ω) | ∥u∥ < k0}.

  2. 𝓙f is strictly convex in Bk0(0).

Proof

  1. Let u ∈ Nf+ (Ω) then ϕu′(1)=0  and  ϕu′′(1)>0. That is, a(u) = b(u) + ∫Ω fu dx and a(u) > (2.2μ∗−1)b(u). Therefore, a(u) = b(u) + ∫Ω fu dx < 1(2.2μ∗−1)a(u) + ∫Ω fu dx. It implies 1−1(2.2μ∗−1)a(u)≤∥f∥H−1∥u∥. So,

    ∥u∥≤(2.2μ∗−1)2(2μ∗−1)∥f∥H−1≤(2.2μ∗−1)2(2μ∗−1)CN,μSH,L2μ∗2.2μ∗−2=12.2μ∗−112(2μ∗−1)SH,L2μ∗2(2μ∗−1)=k0.

  2. By using Hölders inequality and equation (2.2), we have

    ∫Ω∫Ω(u+(x))2μ∗−1(u+(y))2μ∗−1z(x)z(y)|x−y|μ dxdy≤b(u)2μ∗−12μ∗∥z∥NL2≤SH,L−(2μ∗−1)a(u)(2μ∗−1)SH,L−1a(z)=SH,L−2μ∗a(u)(2μ∗−1)a(z). (3.1)

Again using Hölders inequality, Proposition 2.6 and (2.2), we have

∫Ω∫Ω(u+(x))2μ∗(u+(y))2μ∗−2z2(y)|x−y|μ dxdy≤b(u)2μ∗−12μ∗∥z∥NL2≤SH,L−2μ∗a(u)(2μ∗−1)a(z). (3.2)

From equations (3.1), (3.2) and definition of Jf′′ (u)(z, z), we get

Jf′′(u)(z,z)=a(z)−2μ∗∫Ω∫Ω(u+(x))2μ∗−1(u+(y))2μ∗−1z(x)z(y)|x−y|μ dxdy−(2μ∗−1)∫Ω∫Ω(u+(x))2μ∗(u+(y))2μ∗−2z2(y)|x−y|μ dxdy≥a(z)1−2μ∗SH,L−2μ∗a(u)(2μ∗−1)−(2μ∗−1)SH,L−2μ∗a(u)(2μ∗−1)=a(z)1−(2.2μ∗−1)SH,L−2μ∗a(u)(2μ∗−1)>a(z)1−(2.2μ∗−1)(2.2μ∗−1)=0

for u ∈ Bk0(0) ∖ {0}. Then Jf′′ (u) is positive definite for u ∈ Bk0(0) and 𝓙f(u) is strictly convex on Bk0(0).□

Lemma 3.10

It holds that u1 ∈ Nf+ and 𝓙f(u1) = Υf+ (Ω) = Υf(Ω). Moreover, u1 is the unique critical point of 𝓙f in Bk0(0) and u1 is a local minimum of 𝓙f in H01 (Ω).

Proof

Using the proof of [30, Theorem 1.3], we have ∫Ωfu1 dx>0. Now if u1 ∈ Nf− then there exists a unique t−(u1) = 1 > tmax > t+(u1) > 0 such that t+(u1)u1 ∈ Nf+ then by Lemma 3.3 (b) we have

Υf(Ω)≤Υf+(Ω)≤Jf(t+(u1)u1)<Jf(t−(u1)u1)=Jf(u1)=Υf(Ω).

which is a contradiction. It implies u1 ∈ Nf+  and  Υf+(Ω)≤Jf(u1)=Υf(Ω)≤Υf+(Ω) that is, 𝓙f(u1) = Υf(Ω) = Υf+ (Ω). Using Lemma 3.5 and Lemma 3.9, we get u1 is the unique critical point of 𝓙f in Bk0(0) and the proof of local minimum follows from [30, Lemma 3.2].□

Lemma 3.11

Let μ < min{4, N} and u ∈ H01 (Ω) be a critical point of 𝓙f then either u ∈ Nf− or u = u1.

Proof

If u ∈ H01 (Ω) be a critical point of 𝓙f then u ∈ 𝓝f = Nf+∪Nf− . Now using the fact that Nf+∪Nf− = ∅ and Nf+ ⊂ Bk0(0) we have either u ∈ Nf− or u = u1.□

4 Asymptotic estimates and Palais-Smale Analysis

In this section we shall prove that the functional 𝓙f satisfies Palais-Smale condition strictly below the first critical level and (strictly) between the first and second critical levels. To start with, we shall prove several new estimates on the nonlinearity.

It is known from Lemma 2.2 that the best constant SH,L is achieved by the function

u(x)=S(N−μ)(2−N)4(N−μ+2)(C(N,μ))2−N2(N−μ+2)(N(N−2))N−24(1+|x|2)N−22,

which is a solution of the problem −Δu=(|x|−μ∗|u|2μ∗)|u|2μ∗−1 in ℝN with

∫RN|∇u|2 dx=∫RN∫RN|u(x)|2μ∗|u(y)|2μ∗|x−y|μ dxdy=SH,L2N−μN−μ+2.

We may assume R1 = ρ, R2 = 1/ρ for ρ ∈ (0, 12 ). Now, define υρ ∈ Cc∞ (ℝN) such that 0 ≤ υρ(x) ≤ 1 for all x ∈ ℝN, radially symmetric and

υρ(x)=00<|x|<3ρ2,12ρ≤|x|≤12ρ,0|x|≥34ρ,

and

uσϵ(x)=S(N−μ)(2−N)4(N−μ+2)C(N,μ)2−N2(N−μ+2)(N(N−2)ϵ2)N−24(ϵ2+|x−(1−ϵ)σ|2)N−22,

where σ ∈ 𝕊N−1 := {x ∈ ℝN : |x| = 1}, 0 < ϵ ≤ 1. Set

gρϵ,σ(x):=υρ(x)uσϵ(x)∈H01(Ω). (4.1)

Lemma 4.1

  1. a(gρϵ,σ)=b(gρϵ,σ)=SH,L2N−μN−μ+2+oϵ(1) uniformly in σ as ϵ → 0.

  2. J(gρϵ,σ)=N−μ+22(2N−μ)SH,L2N−μN−μ+2+oϵ(1) uniformly in σ as ϵ → 0.

  3. gρϵ,σ ⇀ 0 weakly in H01 (Ω) uniformly in σ as ϵ → 0.

Proof

  1. Observe the fact that there exist constants d1, d2 > 0 such that

    d1<|x−(1−ϵ)σ|<d2 for all x∈B2ρ whenever ϵ<1−2ρ. (4.2)
    ∥∇gρϵ,σ∥L2(RN)−∥∇uϵσ∥L2(RN)≤∫(RN∖B12ρ)∪B2ρ|∇uϵσ|2 dx+ρ−2∫B2ρ|uϵσ|2 dx+ρ2∫B34ρ∖B12ρ|uϵσ|2 dx≤CϵN−2∫(RN∖B12ρ)∪B2ρ|x−(1−ϵ)σ|2|x−(1−ϵ)σ|2N dx+CϵN−2∫B2ρ∪B34ρ∖B12ρdx|x−(1−ϵ)σ|2(N−2)=O(ϵN−2).

    Thus, ∥∇gρϵ,σ∥L2(RN)=∥∇uϵσ∥L2(RN)+oϵ(1)=SH,L2N−μN−μ+2+oϵ(1).

    Next we will prove that b(gρϵ,σ)=SH,L2N−μN−μ+2+oϵ(1) uniformly in σ as ϵ → 0. For this consider

    ∫RN∫RN|gρϵ,σ(x)|2μ∗|gρϵ,σ(y)|2μ∗|x−y|μ dxdy−∫RN∫RN|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy=∫RN∫RN(|υρ(x)|2μ∗|υρ(y)|2μ∗−1)|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy≤C∫B2ρ∫B2ρ+∫B12ρ∖B2ρ∫B2ρ+∫B12ρ∖B2ρ∫RN∖B12ρ+∫RN∖B12ρ∫B2ρ+∫RN∖B12ρ∫RN∖B12ρ|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy,=C∑i=1i=5Ji, (4.3)

    Let ξϵ(x)=ϵN(ϵ2+|x−(1−ϵ)σ|2)N then taking into account the definition of uϵσ , (4.2) and Hardy-Littlewood-Sobolev inequality, we have the following estimates:

    J1≤C(N,μ)∫B2ρS−N(N−μ)2(N−μ+2)C(N,μ)−N(N−μ+2)(N(N−2))N2ξϵ(x) dx2N−μN≤Cϵ2N−μ∫B2ρdx|x−(1−ϵ)σ|2N2N−μN≤Cϵ2N−μ∫B2ρdx2N−μN=O(ϵ2N−μ),J2≤C∫B12ρ∖B2ρξϵ(x) dx2N−μ2N∫B2ρξϵ(x) dx2N−μ2N≤Cϵ2N−μ2∫B2ρdx|x−(1−ϵ)σ|2N2N−μ2N=O(ϵ2N−μ2),J3≤C∫B12ρ∖B2ρξϵ(x)2N−μ2N∫RN∖B12ρξϵ(x) dx2N−μ2N≤Cϵ2N−μ2∫RN∖B12ρdx|x−(1−ϵ)σ|2N2N−μ2N=O(ϵ2N−μ2),
    J4≤C∫RN∖B12ρξϵ(x) dx2N−μ2N∫B2ρξϵ(x) dx2N−μ2N≤Cϵ2N−μ∫RN∖B12ρdx|x−(1−ϵ)σ|2N∫B2ρdx|x−(1−ϵ)σ|2N2N−μ2N=O(ϵ2N−μ),J5≤C∫RN∖B12ρξϵ(x) dx2N−μN≤Cϵ2N−μ∫RN∖B12ρdx|x−(1−ϵ)σ|2N2N−μN=O(ϵ2N−μ).

    Therefore, b(gρϵ,σ)−∫RN∫RN|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy→0 as ϵ → 0 that is, b(gρϵ,σ)→SH,L2N−μN−μ+2 as ϵ → 0 and completes the proof of (i).

  2. Result follows from the definition of 𝓙 and by (i).

  3. Assume by contradiction, gρϵ,σ ⇀ g1 ≢ 0 weakly in H01 (Ω) then gρϵ,σ → g1 strongly in L2(Ω). Then by using the inequality r2(N−2) + s2(N−2) ≤ (r2 + s2)N−2 for all r, s ≥ 0, we have

    0≤∫Ω|gρϵ,σ|2 dx≤C∫3ρ2≤|x|≤34ρϵN−2(ϵ2+|x−(1−ϵ)σ|2)N−2 dx=C∫3ρ2≤|y+(1−ϵ)σ|≤34ρϵN−2ϵ2(N−2)+|y|2(N−2) dy≤C∫034ρ+(1−ϵ)ϵN−2rN−1ϵ2(N−2)+r2(N−2) dy→0.

It yields a contradiction. Hence results follows.□

Lemma 4.2

Let σ ∈ 𝕊N−1 and ϵ ∈ (0, 1), then the following holds:

  1. limρ→0supσ∈SN−1,ϵ∈(0,1]∥∇(gρϵ,σ−uϵσ)∥L2(RN)2=0.

  2. limρ→0supσ∈SN−1,ϵ∈(0,1]∥gρϵ,σ∥NL2.2μ∗=∥uϵσ∥NL2.2μ∗.

Proof

  1. Consider

    ∫RN|∇gρϵ,σ−∇uϵσ|2dx≤2∫RN|uϵσ(x)∇υρ(x)|2 dx+2∫RN|∇uϵσ(x)υρ(x)−∇uϵσ(x)|2 dx≤Cρ−2∫B2ρ|uϵσ(x)|2 dx+∫B2ρ|∇uϵσ(x)|2 dx+Cρ2∫B34ρ∖B12ρ|uϵσ(x)|2 dx+∫RN∖B12ρ|∇uϵσ(x)|2 dx. (4.4)

    From the definition of uϵσ , we have the following estimates

    ρ−2∫B2ρ|uϵσ(x)|2 dx≤Cρ−2∫B2ρ dx≤CρN−2,∫B2ρ|∇uϵσ(x)|2 dx≤C∫B2ρ|x−tσ| dx≤C∫B2ρ dx≤CρN,ρ2∫B34ρ∖B12ρ|uϵσ(x)|2 dx≤Cρ2∫B34ρ∖B12ρ1|x|2N−4 dx≤CρN−2,∫RN∖B12ρ|∇uϵσ(x)|2 dx≤C∫RN∖B12ρ1|x|2N−2 dx≤CρN−2.

    Therefore, from above estimates and (4.4), we obtain desired result.

  2. Consider

    ∥gρϵ,σ∥NL2.2μ∗−∥uϵσ∥NL2.2μ∗=∫RN∫RN(υρ2μ∗(x)υρ2μ∗(y)−1)|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy≤C∑i=15Ji,

    where Ji are defined in (4.3). Using the Hardy-Littlewood-Sobolev inequality and the definition of ξϵ, we have the following estimates:

    J1≤C(N,μ)∫B2ρξϵ(x) dx2N−μN≤C∫B2ρdx2N−μN≤Cρ2N−μ,J2≤C(N,μ)∫B12ρ∖B2ρξϵ(x) dx2N−μ2N∫B2ρξϵ(x) dx2N−μ2N≤C∫B2ρdx2N−μN≤Cρ2N−μ2,
    J3≤C(N,μ)∫B12ρ∖B2ρξϵ(x)dx2N−μ2N∫RN∖B12ρξϵ(x)dx2N−μ2N≤C∫RN∖B12ρdx|x−(1−ϵ)σ|2N2N−μ2N=∫|y+(1−ϵ)σ|≥12ρdy|y|2N2N−μ2N≤∫|y|≥12ρ−1dy|y|2N2N−μ2N≤C(2ρ)N1−(2ρ)N2N−μ2N,

Now using the same estimates as above we can easily obtain

J4≤Cρ2N−μ2 and J5≤C(2ρ)N1−(2ρ)N2N−μN.

Hence supσ∈SN−1,ϵ∈(0,1]∥gρϵ,σ∥NL2.2μ∗−∥uϵσ∥NL2.2μ∗→0 as ρ → 0 and completes the proof.□

Lemma 4.3

The following asymptic estimates hold:

  1. a(gρϵ,σ)≤SH,L2N−μN−μ+2+O(ϵN−2).

  2. b(gρϵ,σ)≤SH,L2N−μN−μ+2+O(ϵN).

  3. b(gρϵ,σ)≥SH,L2N−μN−μ+2−O(ϵ2N−μ2).

Proof

Part (i) follows from Lemma 4.1 (i). For part (ii) we will first estimate the integral ∫Ω|gρϵ,σ|2∗ dx. Since

∫Ω|gρϵ,σ|2∗ dx≤C∫B34ρ∖B3ρ2|uϵσ|2∗ dx≤∫B34ρ∖B12ρ|uϵσ|2∗ dx+∫B12ρ∖B3ρ2|uϵσ|2∗ dx

and

∫B34ρ∖B12ρ|uϵσ|2∗ dx≤CϵN∫B34ρ∖B12ρdx|x−(1−ϵ)σ|2N=O(ϵN),∫B12ρ∖B3ρ2|uϵσ|2∗ dx≤∫RN|uϵσ|2∗ dx=SNN−μ+2C(N,μ)−NN−μ+2.

It implies ∫Ω|gρϵ,σ|2∗ dx≤SNN−μ+2C(N,μ)−NN−μ+2+O(ϵN) and now using this and Hardy-Littlewood-Sobolev inequality we have

b(gρϵ,σ)=∫Ω∫Ω|gρϵ,σ(x)|2μ∗|gρϵ,σ(y)|2μ∗|x−y|μ dxdy≤C(N,μ)∫Ω|gρϵ,σ|2∗ dx2N−μN≤C(N,μ)SNN−μ+2C(N,μ)−NN−μ+2+O(ϵN)2N−μN≤SH,L2N−μN−μ+2+O(ϵN).

This proves part (ii). Now to prove part (iii), consider

b(gρϵ,σ)=∫Ω∫Ω|gρϵ,σ(x)|2μ∗|gρϵ,σ(y)|2μ∗|x−y|μ dxdy≥∫B12ρ∖B2ρ∫B12ρ∖B2ρ|gρϵ,σ(x)|2μ∗|gρϵ,σ(y)|2μ∗|x−y|μ dxdy=∫RN∫RN|uϵσ(x)|2μ∗|uϵσ(y)|2μ∗|x−y|μ dxdy−∑i=1i=5Ji,

where Ji are defined in equation (4.3). Using the proof of Lemma 4.1(i) and the fact that ∥uϵσ∥NL2.2μ∗=SH,L2N−μN−μ+2 + oϵ(1), we obtain the required result.□

Now we will give a Lemma which is taken from [18]. For the sake of completeness, we provide a complete proof.

Lemma 4.4

If μ < min{4, N} then

b(u1+tgρϵ,σ)≥b(u1)+b(tgρϵ,σ)+C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy+2.2μ∗t∫Ω∫Ω(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy−O(ϵ(2N−μ4)Θ)  for all  Θ<1,

where u1 is the local minimum obtained in Lemma 3.10.

Proof

We will divide the proof in two cases:

  1. 2μ∗ > 3.

    It is easy to see that there exists  > 0 such that

    (a+b)p≥ap+bp+pap−1b+A^abp−1 for all a,b≥0 and p>3,

    which implies that

    b(u1+tgρϵ,σ)≥b(u1)+b(tgρϵ,σ)+C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy+2.2μ∗t∫Ω∫Ω(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy, where C^=min{A^,2.2μ∗}.

  2. 2 < 2μ∗ ≤ 3.

    We recall the inequality from [7, Lemma 4]: there exist C(depending on 2μ∗ ) such that, for all a, b ≥ 0,

    (a+b)2μ∗≥a2μ∗+b2μ∗+2μ∗a2μ∗−1b+2μ∗ab2μ∗−1−Cab2μ∗−1 if a≥b,a2μ∗+b2μ∗+2μ∗a2μ∗−1b+2μ∗ab2μ∗−1−Ca2μ∗−1b if a≤b, (4.5)

    Consider Ω × Ω = O1 ∪ O2 ∪ O3 ∪ O4, where

    O1={(x,y)∈Ω×Ω∣u1(x)≥tgρϵ,σ(x) and u1(y)≥tgρϵ,σ(y)},O2={(x,y)∈Ω×Ω∣u1(x)≥tgρϵ,σ(x) and u1(y)<tgρϵ,σ(y)},O3={(x,y)∈Ω×Ω∣u1(x)<tgρϵ,σ(x) and u1(y)≥tgρϵ,σ(y)},O4={(x,y)∈Ω×Ω∣u1(x)<tgρϵ,σ(x) and u1(y)<tgρϵ,σ(y)}.

    Also, define the b(u)|Oi=∫∫Oi(u(x))2μ∗(u(y))2μ∗|x−y|μ dxdy , for all u ∈ H01 (Ω) and i = 1, 2, 3, 4.

  3. when (x, y) ∈ O1.

    Employing (4.5), we have the following inequality:

    b(u1+tgρϵ,σ)|O1≥(b(u1)+b(tgρϵ,σ))|O1+2.2μ∗t2.2μ∗−1∫∫O1(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μdxdy+2.2μ∗t∫∫O1(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μdxdy−Aϵ1,

    where Aϵ1 is sum of eight non-negative integrals and each integral has an upper bound of the form C∫∫O1u1(x)(tgρϵ,σ(x))2μ∗−1(u1(y))2μ∗|x−y|μ dxdy  or  C∫∫O1u1(y)(tgρϵ,σ(y))2μ∗−1(u1(x))2μ∗|x−y|μ dxdy. Write (tgρϵ,σ(x))2μ∗−1 = (tgρϵ,σ(x))r.(tgρϵ,σ(x))s with 2μ∗−1=r+s,0<s<2μ∗2. Then utilizing the definition of O1, u1 ∈ L∞(Ω) and Hardy-Littlewood-Sobolev inequality, we have

    ∫∫O1u1(x)(tgρϵ,σ(x))2μ∗−1(u1(y))2μ∗|x−y|μ dxdy≤C∫∫O1(u1(x))1+r(tgρϵ,σ(x))s(u1(y))2μ∗|x−y|μ dxdy≤C∫Ω∫Ω(tgρϵ,σ(x))s(u1(y))2μ∗|x−y|μ dxdy≤C∫Ω∫Ωϵs(N−2)2|x−y|μ|x−(1−ϵ)σ|s(N−2) dxdy≤Cϵs(N−2)2∫Ωdx|x−(1−ϵ)σ|s(2N)(N−2)2N−μ2N−μ2N≤Cϵs(N−2)2∫Ωdx|x−(1−ϵ)σ|s(2N)(N−2)2N−μ2N−μ2N.

    By the choice of s we have ∫Ωdx|x−(1−ϵ)σ|s(2N)(N−2)2N−μ<∞. As a result, we get

    ∫∫O1u1(x)(tgρϵ,σ(x))2μ∗−1(u1(y))2μ∗|x−y|μ dxdy≤O(ϵ(2N−μ4)Θ) for all Θ<1.

    In a similar manner, we have

    C∫∫O1u1(y)(tgρϵ,σ(y))2μ∗−1(u1(x))2μ∗|x−y|μ dxdy≤O(ϵ(2N−μ4)Θ) for all Θ<1.

  4. when (x, y) ∈ O2.

    Once again using (4.5), we have the following inequality:

    b(u1+tgρϵ,σ)|O2≥[b(u1)+b(tgρϵ,σ)]|O2+2.2μ∗t2.2μ∗−1∬O2(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μdxdy+2.2μ∗t∬O2(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy−Aϵ2,

    where Aϵ2 is sum of eight non-negative integrals and each integral has an upper bound of the form C∫∫O2u1(x)(tgρϵ,σ(x))2μ∗−1(gρϵ,σ(y))2μ∗|x−y|μ dxdy or C∫∫O2(u1(y))2μ∗−1(tgρϵ,σ(y))(u1(x))2μ∗|x−y|μ dxdy. By the similar estimates as in Subcase 1, definition of O2, the fact that tgρϵ,σ∈H01(Ω) and regularity of u1, we have

    ∫∫O2u1(x)(tgρϵ,σ(x))2μ∗−1(gρϵ,σ(y))2μ∗|x−y|μ dxdy≤O(ϵ(2N−μ4)Θ)  for all  Θ<1.

    Write (u1(y))2μ∗−1=(u1(y))r.(u1(y))s with 2μ∗−1=r+s,0<1+s<2μ∗2. Then utilizing the definition of O2, u1 ∈ L∞(Ω) and Hardy-Littlewood-Sobolev inequality, we have

    ∫∫O2(u1(y))2μ∗−1(tgρϵ,σ(y))(u1(x))2μ∗|x−y|μ dxdy≤∫∫O2(u1(y))r(tgρϵ,σ(y))1+s(u1(x))2μ∗|x−y|μ dxdy≤C∫Ω∫Ω(tgρϵ,σ(y))1+s(u1(x))2μ∗|x−y|μ dxdy≤C∫Ω∫Ωϵ(1+s)(N−2)2|x−y|μ|y−(1−ϵ)σ|(1+s)(N−2) dxdy≤Cϵ(1+s)(N−2)2∫Ωdy|y−(1−ϵ)σ|(1+s)(2N)(N−2)2N−μ2N−μ2N≤Cϵ(1+s)(N−2)2∫Ωdy|y−(1−ϵ)σ|(1+s)(2N)(N−2)2N−μ2N−μ2N.

    By the choice of s we have ∫Ωdx|x−(1−ϵ)σ|(1+s)(2N)(N−2)2N−μ<∞. Hence we obtain

    ∫∫O2(u1(y))2μ∗−1(tgρϵ,σ(y))(u1(x))2μ∗|x−y|μ dxdy≤O(ϵ(2N−μ4)Θ)  for all  Θ<1.

  5. when (x, y) ∈ O3.

    Using (4.5), we have

    b(u1+tgρϵ,σ)|O3≥(b(u1)+b(tgρϵ,σ))|O3+2.2μ∗t2.2μ∗−1∬O3(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μdxdy+2.2μ∗t∬O3(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy−Aϵ3,

    where Aϵ3 is sum of eight non-negative integrals and each integral has an upper bound of the form C∬O3(u1(x))2μ∗−1(tgρϵ,σ(x))(u1(y))2μ∗|x−y|μ dxdy or C∬O3u1(y)(tgρϵ,σ(y))2μ∗−1(gρϵ,σ(x))2μ∗|x−y|μ dxdy. By the similar estimates as in Subcase 1, Subcase 2, definition of O3 and regularity of u1, we get Aϵ3≤O(ϵ(2N−μ4)Θ) for all Θ < 1.

  6. when (x, y) ∈ O4.

    Using (4.5), we have

    b(u1+tgρϵ,σ)|O4≥(b(u1)+b(tgρϵ,σ))|O4+2.2μ∗t2.2μ∗−1∬O4(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μdxdy+2.2μ∗t∬O4(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy−Aϵ4,

    where Aϵ4 is sum of eight non-negative integrals and each integral has an upper bound of the form C∫∫O4(u1(x))2μ∗−1(tgρϵ,σ(x))(tgρϵ,σ(y))2μ∗|x−y|μ dxdy or C∫∫O4u1(y)(tgρϵ,σ(y))2μ∗−1(gρϵ,σ(x))2μ∗|x−y|μ dxdy. By the similar estimates as in Subcase 2, we have

    Aϵ4≤O(ϵ(2N−μ4)Θ) for all Θ<1.

    From all subcases we obtain Aϵi≤O(ϵ(2N−μ4)Θ) for all Θ < 1 and i = 1, 2, 3, 4. Combining all sub cases we conclude Case 2. From Case 1 and Case 2 we have the required result.□

Proposition 4.5

Let μ < min{4, N} then there exists ϵ0 > 0 such that for every 0 < ϵ < ϵ0 we have

supt≥0Jf(u1+tgρϵ,σ)<Jf(u1)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 uniformly in σ∈SN−1,

where u1 is the local minimum in Lemma 3.10.

Proof

By Lemma 3.8, u ∈ L∞(Ω) and u > 0 in Ω. This implies

b(u1+tgρϵ,σ)=∫Ω∫Ω(u1+tgρϵ,σ(x))2μ∗(u1+tgρϵ,σ(y))2μ∗|x−y|μ dxdy.

Claim 1: There exists a R0 > 0 such that

I=∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy≥C^R0ϵN−22.

Clearly,

I≥∫B12ρ∖B2ρ∫B12ρ∖B2ρ(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy≥C∫B12ρ∖B2ρ∫B12ρ∖B2ρ(uϵσ(x))2μ∗(uϵσ(y))2μ∗−1|x−y|μ dxdy≥C∫B12ρ∖B2ρ∫B12ρ∖B2ρϵ3N2+1−μ dxdy|x−y|μ(ϵ2+|x−(1−ϵ)σ|2)2N−μ2(ϵ2+|y−(1−ϵ)σ|2)N−μ+22.

For any ϵ < 1 – 2ρ there exists c > 0 such that 1 – ϵ > c > 2ρ so we get

I≥Cϵ3N2+1−μ∫Bc∫Bcdzdw|z−w|μ(ϵ2+|z|2)2N−μ2(ϵ2+|w|2)N−μ+22≥CϵN−22∫Bc∫Bcdzdw|z−w|μ(1+|z|2)2N−μ2(1+|w|2)N−μ+22=O(ϵN−22).

This proves the claim 1. Now using Lemma 4.4, we have

Jf(u1+tgρϵ,σ)≤12a(u1)+12a(tgρϵ,σ)+t〈u1,gρϵ,σ〉H01(Ω)−12.2μ∗b(u1)−12.2μ∗b(tgρϵ,σ)−C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy−∫Ωfu1 dx−t∫Ωfgρϵ,σ dx−t∫Ω∫Ω(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy+O(ϵ(2N−μ4)Θ).

for all Θ < 1. Taking Θ=22μ∗, we have

Jf(u1+tgρϵ,σ)≤12a(u1)+12a(tgρϵ,σ)+t〈u1,gρϵ,σ〉H01(Ω)−12.2μ∗b(u1)−12.2μ∗b(tgρϵ,σ)−C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1u1(y)|x−y|μ dxdy−∫Ωfu1 dx−t∫Ωfgρϵ,σ dx−t∫Ω∫Ω(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy+o(ϵN−22).

This on utilizing Lemma 4.3 and claim 1 gives

Jf(u1+tgρϵ,σ)≤12a(u1)+12a(tgρϵ,σ)+t〈u1,gρϵ,σ〉H01(Ω)−12.2μ∗b(u1)−12.2μ∗b(tgρϵ,σ)−C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1|x−y|μ dxdy−∫Ωfu1 dx−t∫Ωfgρϵ,σ dx−t∫Ω∫Ω(u1(x))2μ∗(u1(y))2μ∗−1gρϵ,σ(y)|x−y|μ dxdy+o(ϵN−22)=Jf(u1)+J(tgρϵ,σ)−C^t2.2μ∗−1∫Ω∫Ω(gρϵ,σ(x))2μ∗(gρϵ,σ(y))2μ∗−1|x−y|μ dxdy+o(ϵN−22)≤Jf(u1)+t22SH,L2N−μN−μ+2+O(ϵN−2)−t2.2μ∗2.2μ∗SH,L2N−μN−μ+2−O(ϵ2N−μ2)−t2.2μ∗−1C^R0ϵN−22+o(ϵN−22).

Now define K(t):=t22SH,L2N−μN−μ+2+O(ϵN−2)−t2.2μ∗2.2μ∗SH,L2N−μN−μ+2−O(ϵ2N−μ2)−t2.2μ∗−1C^R0ϵN−22 then K(t) → ∞ as t → ∞ and limt→0+ K(t) > 0 so there exists a tϵ > 0 such that supt>0 K(t) is attained and tϵ<SH,L2N−μN−μ+2+O(ϵN−2)SH,L2N−μN−μ+2−O(ϵ2N−μ2)12.2μ∗−2 := SH,L(ϵ). Moreover there exists a t1 > 0 such that for sufficiently small ϵ > 0 we have tϵ > t1. Clearly the function

t↦t22SH,L2N−μN−μ+2+O(ϵN−2)−t2.2μ∗2.2μ∗SH,L2N−μN−μ+2−O(ϵ2N−μ2)

is an increasing function in [0, SH,L(ϵ)]. Therefore,

supt≥0Jf(u1+tgρϵ,σ)≤Jf(u1)+N−μ+22(2N−μ)SH,L2N−μN−μ+2+O(ϵmin{2N−μ2,N−2})−t12.2μ∗−1C^R0ϵN−22+o(ϵN−22).

Hence there exits a ϵ0 > 0 such that for every 0 < ϵ < ϵ0 we have

supt≥0Jf(u1+tgρϵ,σ)<Jf(u1)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 uniformly in σ∈SN−1.

□

Lemma 4.6

The following holds:

  1. H01(Ω)∖Nf−=U1∪U2, where

    U1:=u∈H01(Ω)∖{0}|u+≢0,∥u∥<t−u∥u∥∪{0},U2:=u∈H01(Ω)∖{0}|u+≢0,∥u∥>t−u∥u∥.

  2. Nf+ ⊂ U1.

  3. For each 0 < ϵ ≤ ϵ0, there exists t0 > 1 and such that u1 + t0gρϵ,σ ∈ U2.

  4. For each 0 < ϵ < ϵ0, there exists s0 ⊂ (0, 1) and such that u1+s0t0gρϵ,σ∈Nf−.

  5. Υf−<Υf+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Proof

  1. It holds by Lemma 3.3 (d).

  2. Let u ∈ Nf+ then t+(u) = 1 and 1<t+(u)<tmax<t−(u)=1∥u∥t−u∥u∥ that is, Nf+ ⊂ U1.

  3. First, we will show that there exists a constant c > 0 such that 0<t−u1+tgρϵ,σ∥u1+tgρϵ,σ∥<c for all t > 0. On the contrary, let there exist a sequence {tn} such that tn → ∞ and t−u1+tngρϵ,σ∥u1+tngρϵ,σ∥→∞ as n → ∞. Define un:=u1+tngρϵ,σ∥u1+tngρϵ,σ∥ so there exists t–(un) such that t–(un)un ∈ Nf− . By dominated convergence theorem,

    b(un)=b(u1+tngρϵ,σ)∥u1+tngρϵ,σ∥2.2μ∗=b(u1tn+gρϵ,σ)∥u1tn+gρϵ,σ∥2.2μ∗→b(gρϵ,σ)∥gρϵ,σ∥2.2μ∗ as n→∞.

    Hence, 𝓙f(t–(un)un) → –∞ as n → ∞, contradicts the fact that 𝓙f is bounded below on 𝓝f. Therefore, there exists c > 0 such that 0<t−u1+tgρϵ,σ∥u1+tgρϵ,σ∥<c for all t > 0. Let t0=|c2−∥u1∥2|12∥gρϵ,σ∥+1 then

    ∥u1+t0gρϵ,σ∥2=∥u1∥2+t02∥gρϵ,σ∥2+2t0〈u1,gρϵ,σ〉≥∥u1∥2+|c2−∥u1∥2|≥c2≥t−u1+tgρϵ,σ∥u1+tgρϵ,σ∥2.

    It implies that u1 + t0gρϵ,σ ∈ U2.

  4. For each 0 < ϵ < ϵ0, define a path ξϵ(s) = u1 + st0gρϵ,σ for s ∈ [0, 1]. Then

    ξϵ(0)=u1andξϵ(1)=u1+t0gρϵ,σ∈U2.

    Since 1∥u∥t−u∥u∥ is a continuous function and ξϵ([0, 1]) is connected. So, there exists s0 ∈ [0, 1] such that ξϵ(s0)=u1+s0t0gρϵ,σ∈Nf−.

  5. Using part (iv) and Proposition 4.5.□

At this point we will state Global compactness Lemma for the functional 𝓙f which is a version of Theorem 4.4 of [19].

Lemma 4.7

Let {un} ⊂ H01 (Ω) be such that 𝓙f(un) → c, Jf′ (un) → 0. Then passing if necessary to a subsequence, there exists a solution v0 ∈ H01 (Ω) of

−Δu=∫Ω|u+(y)|2μ∗|x−y|μdy|u+|2μ∗−1+f in Ω

and (possibly) k ∈ ℕ ∪ {0}, non-trivial solutions {v1, v2, …, vk} of

−Δu=(|x|−μ∗|u+|2μ∗)|u+|2μ∗−1 in RN

with vi ∈ D1,2(ℝN) and k sequences {yni}n ⊂ Ω and {λni}n ⊂ ℝ+ i = 1, 2, … k, satisfying

1λnidist(yni,∂Ω)→∞, and ∥un−v0−∑i=1k(λni)2−N2vi((.−yni)/λni)∥D1,2(RN)→0,n→∞,∥un∥D1,2(RN)2→∑i=0k∥vi∥D1,2(RN)2, as n→∞,Jf(v0)+∑i=1kJ∞(vi)=c,

where J∞(u):=12∫RN|∇u|2 dx−12.2μ∗∫RN∫RN|u+(x)|2μ∗|u+(y)|2μ∗|x−y|μ dxdy,u∈D1,2(RN).

Lemma 4.8

  1. Let {un} be a (PS)c sequence for 𝓙f with c<Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 then there exists a subsequence still denoted by {un} and a nonzero u0 ∈ H01 (Ω) such that un → u0 strongly in H01 (Ω) and 𝓙f(u0) = c.

  2. Let {un} ⊂ Nf− be a (PS)c sequence for 𝓙f with

    Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2<c<Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2

    then there exists subsequence still denoted by {un} and a nonzero u0 ∈ Nf− such that un → u0 strongly in H01 (Ω) and 𝓙f(u0) = c.

Proof

Proof of (i) follows from [30, Lemma 3.4]. To prove (ii), Let {un} be a (PS)c sequence then by standard arguments {un} is bounded in H01 (Ω) and there exists a subsequence of {un} still denoted by {un} and u0 ∈ H01 (Ω) such that un ⇀ u0 in H01 (Ω) and Jf′ (u0) = 0. Then by Lemma 3.11, we have either u0 ∈ Nf− or u0 = u1. Now using Lemma 4.7 we obtain

Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2≥c=Jf(u0)+∑i=1kJ∞(vi)≥Υf(Ω)+kN−μ+22(2N−μ)SH,L2N−μN−μ+2.

which on using Lemma 4.6(e), gives k ≤ 1. By [19, corollary 3.3], we get v1 is a constant multiple of Talenti function that is, J∞(v1)=N−μ+22(2N−μ)SH,L2N−μN−μ+2. If k = 0 then we are done and if k = 1 and u0 = u1, then

c=Jf(u0)+N−μ+22(2N−μ)SH,L2N−μN−μ+2=Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2,

a contradiction. If k = 1 and u0 ∈ Nf− , we get

c=Jf(u0)+N−μ+22(2N−μ)SH,L2N−μN−μ+2≥Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2,

which is again a contradiction. Hence k = 0 and result follows.□

5 Existence of Second and third Solution

In this section we will show the existence of second and third solution of problem (Pf). To prove this, we shall show that for a sufficiently small δ > 0,

cat({u∈Nf−:Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2−δ})≥2,

where cat(X) is the category of the set X is defined in the Definition 5.1. And then employing Lemma 5.2, we conclude the existence of second and third solutions. We shall first gather some preliminaries.

For c ∈ ℝ, we define

bc(u)=cb(u),Jc(u)=12a(u)−12.2μ∗bc(u),Mc={u∈H01(Ω)∖{0}|〈Jc′(u),u〉=0}.

We denote

[Jf≤c]={u∈Nf−|Jf(u)≤c}.

Definition 5.1

  1. For a topological space X, we say that a non-empty, closed subset Y ⊂ X is contractible to a point in X if and only if there exists a continuous mapping ϱ : [0, 1] × Y → X such that for some x0 ∈ X, ϱ(0, x) = x for all x ∈ Y and ϱ(1, x) = x0 for all x ∈ Y.

  2. We define

    cat(X)=min{k∈N| there exists closed subsets Y1,Y2,⋯Yk⊂X such that Yj is contractible to a point in X for all j and ∪j=1kYj=X}

Lemma 5.2

[2] Suppose that X is a Hilbert manifold and G ∈ C1(X, ℝ). Assume that there are c1 ∈ ℝ and k ∈ ℕ, such that

  1. G satisfies the Palais-Smale condition for energy level c ≤ c1;

  2. cat({x ∈ X | G(x) ≤ c1}) ≥ k.

Then G has at least k critical points in {x ∈ X | G(x) ≤ c1}.□

Lemma 5.3

[1, Theorem 2.5] Let X be a topological space. Suppose that there are two continuous maps Φ : 𝕊N–1 → X and Ψ : X → 𝕊N–1 such that ΨoΦ is homotopic to the identity map of 𝕊N–1. Then cat(X) ≥ 2.□

Now we will proof a Lemma which will relate the functional 𝓙f and 𝓘c. Note that for each u ∈ H01 (Ω) there exists a unique t– > 0 and a unique t* > 0 such that t– u ∈ Nf− and t* u ∈ 𝓝.

Lemma 5.4

  1. For each u ∈ Σ := {u ∈ H01 (Ω)| ∥u∥ = 1}, there exists a unique tc(u) > 0 such that tc(u)u ∈ 𝓜c and

    maxt≥0Jc(tu)=Jc(tc(u)u)=N−μ+22(2N−μ)bc(u)−N−2N−μ+2.

  2. For each u ∈ H01 (Ω) with u+ ≢ 0 and 0 < ω < 1, we have

    (1−ω)J11−ω(u)−12ω∥f∥H−12≤Jf(u)≤(1+ω)J11+ω(u)+12ω∥f∥H−12

  3. For each u ∈ Σ and 0 < ω < 1, we have

    (1−ω)2N−μN−μ+2J(t∗u)−12ω∥f∥H−12≤Jf(t−u)≤(1+ω)2N−μN−μ+2J(t∗u)+12ω∥f∥H−12.

  4. There exists e11 > 0 such that if 0 < ∥f∥H–1 < e11 then Υf−>0.

Proof

  1. For each u ∈ Σ, define k(t)=12t2−t2.2μ∗2.2μ∗bc(u), then if

    tc(u)=1bc(u)12(2μ∗−1),

    we obtain k′(tc(u)) = 0 and k″(tc(u)) < 0. Therefore, there exists a unique tc(u) > 0 such that

    maxt≥0Jc(tu)=Jc(tc(u)u)=N−μ+22(2N−μ)bc(u)−N−2N−μ+2.

  2. For 0 < ω < 1, we have

    |∫Ωfu dx|≤∥f∥H−1∥u∥≤ω2∥u∥2+12ω∥f∥H−12,

    and for u ∈ H01 (Ω) with u+ ≢ 0 by the above inequality, we get

    1−ω2∥u∥2−12.2μ∗b(u)−12ω∥f∥H−12≤Jf(u)≤1+ω2∥u∥2−12.2μ∗b(u)+12ω∥f∥H−12.

    This implies that

    (1−ω)J11−ω(u)−12ω∥f∥H−12≤Jf(u)≤(1+ω)J11+ω(u)+12ω∥f∥H−12.

  3. Using part (ii), we obtain the following estimate for each u ∈ Σ and 0 < ω < 1

    (1−ω)J11−ω(t11−ω(u)u)−12ω∥f∥H−12≤Jf(t−(u)u)≤(1+ω)J11+ω(t11+ω(u)u)+12ω∥f∥H−12. (5.1)

    Using (5.1) in part (i) we get

    J11−ω(t11−ω(u)u)=N−μ+22(2N−μ)b11−ω(u)−N−2N−μ+2=(1−ω)N−2N−μ+2N−μ+22(2N−μ)b(u)−N−2N−μ+2=(1−ω)N−2N−μ+2J(t∗u).

    Therefore, we get

    (1−ω)2N−μN−μ+2J(t∗u)−12ω∥f∥H−12≤Jf(t−u)≤(1+ω)2N−μN−μ+2J(t∗u)−12ω∥f∥H−12.

  4. Combining part (iii) with the fact that Υ0=N−μ+22(2N−μ)SH,L2N−μN−μ+2>0 contributes that

    Υf−(Ω)>(1−ω)2N−μN−μ+2Υ0−12ω∥f∥H−12=(1−ω)2N−μN−μ+2N−μ+22(2N−μ)SH,L2N−μN−μ+2−12ω∥f∥H−12.

    Thus, there exists e11 > 0 such that Υf− (Ω) > 0 whenever ∥f∥H–1 < e11□

Lemma 5.5

If Ω satisfies condition (A) then there exists a δ0 > 0 such that if u ∈ 𝓝 with J(u)≤N−μ+22(2N−μ)SH,L2N−μN−μ+2+ δ0, then ∫RNx|x||∇u|2 dx≠0

Proof

Let {un} ∈ 𝓝 such that J(un)=N−μ+22(2N−μ)SH,L2N−μN−μ+2+o(1) and ∫RNx|x||∇un|2 dx=0. Since {un} ∈ 𝓝 therefore by Lemma 2.9, {un} is a Palais-Smale sequence of 𝓙 at level N−μ+22(2N−μ)SH,L2N−μN−μ+2. Now using [19, Theorem 4.4] and Remark 2.8, we have

∥un−(λn1)2−N2v1((.−yn1)/λn1)∥D1,2(RN)→0,

where v1 is a minimizer of SH,L, λn1 ∈ ℝ+, yn1 ∈ Ω. Moreover, if n → ∞ then λn1 → 0, yn1|yn1|→y0 is the unit vector in ℝN. Thus we obtain

0=∫RNx|x||∇un|2 dx=∫RNx|x||∇un|2−|∇(λn1)2−N2v1((.−yn1)/λn1)|2 dx+∫RNx|x||∇(λn1)2−N2v1((.−yn1)/λn1)|2 dx=on(1)+∫RNyn1+λn1z|yn1+λn1z||∇v1(z)|2 dz=on(1)+y0SH,L2N−μN−μ+2,

as n → ∞, which is not possible.□

For 0 < ϵ ≤ ϵ0 (defined in Proposition 4.5), define Hϵ : 𝕊N–1 → H01 (Ω) as

Hϵ(σ)=u1+s0t0gρϵ,σ,

where the function u1 + s0 t0gρϵ,σ defined in Lemma 4.6.

Lemma 5.6

There exists a δϵ ∈ ℝ+ such that

Hϵ(SN−1)⊂Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2−δϵ.

Proof

Trivially, Hϵ(σ)=u1+s0t0gρϵ,σ∈Nf−. So we only have to prove that 𝓙f(u1 + s0 t0gρϵ,σ ) ≤ Υf(Ω) + N−μ+22(2N−μ)SH,L2N−μN−μ+2−δϵ for some δϵ > 0. Since by Proposition 4.5,

supt≥0Jf(u1+tgρϵ,σ)<Jf(u1)+N−μ+22(2N−μ)SH,L2N−μN−μ+2=Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Hence there exists a δϵ > 0 such that

Jf(u1+s0t0gρϵ,σ)≤supt≥0Jf(u1+tgρϵ,σ)≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2−δϵ.

□

Lemma 5.7

There exists a e22 > 0 such that ∥f∥H–1 < e22 then for any

u∈Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2we have∫RNx|x||∇u|2 dx≠0.

Proof

Let u∈Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 then Jf(u)≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 and u ∈ Nf− , that is, 1∥u∥t−u∥u∥=1. Since Υf(Ω) < 0 we have Jf(u)≤N−μ+22(2N−μ)SH,L2N−μN−μ+2. So for u∥u∥∈Σ there exits a t* > 0 such that t∗u∥u∥∈N which on using Lemma 5.4 (iii) implies

(1−ω)2N−μN−μ+2Jt∗u∥u∥−12ω∥f∥H−12≤Jft−u∥u∥=Jf(u).

Now using Lemma 3.4, we have

Jt∗u∥u∥≤(1−ω)−2N−μN−μ+2Jf(u)+12ω∥f∥H−12≤(1−ω)−2N−μN−μ+2N−μ+22(2N−μ)SH,L2N−μN−μ+2+12ω∥f∥H−12=(1−ω)−2N−μN−μ+2−1N−μ+22(2N−μ)SH,L2N−μN−μ+2+N−μ+22(2N−μ)SH,L2N−μN−μ+2+12ω(1−ω)2N−μN−μ+2∥f∥H−12.

Choose ω0 > 0 such that for 0 < ω < ω0, we have (1−ω)−2N−μN−μ+2−1N−μ+22(2N−μ)SH,L2N−μN−μ+2<δ02 where δ0 is defined in Lemma 5.5. Now for 0 < ω < ω0 choose e22 such that if ∥f∥H–1 < e22 then 12ω(1−ω)2N−μN−μ+2∥f∥H−12<δ02. Therefore, we obtain

Jt∗u∥u∥≤N−μ+22(2N−μ)SH,L2N−μN−μ+2+δ0

Using Lemma 5.5 we conclude the result.□

Define G:[Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2]→SN−1 by

G(u)=∫RNx|x||∇u|2 dx|∫RNx|x||∇u|2 dx|.

Note that from Lemma 5.5, G is well defined.

Lemma 5.8

For 0 < ϵ < ϵ0 and ∥f∥H–1 < e22, the map

GoHϵ:SN−1→SN−1

is homotopic to the identity.

Proof

Define K:={u∈H01(Ω)∖{0}|∫RNx|x||∇u|2 dx≠0} and G : 𝓚 → 𝕊N–1 by

G¯(u)=∫RNx|x||∇u|2 dx/|∫RNx|x||∇u|2 dx|

as an extension of G. This on using Lemma 4.1 and Lemma 5.5, gives ∫RNx|x||∇gρϵ,σ|2 dx≠0 for sufficiently small ϵ. Thus, G¯(gρϵ,σ) is well defined. Now let y : [s1, s2] → 𝕊N–1 be a regular geodesic between G¯(gρϵ,σ) and G(Hϵ(σ)) such that y(s1) = G¯(gρϵ,σ) and y(s2) = G(Hϵ(σ)). Moreover, by a analogous argument as in Lemma 4.1, for δ0 > 0 there exists a ϵ0 > 0 such that

J(gρ2(1−λ)ϵ)<N−μ+22(2N−μ)SH,L2N−μN−μ+2+δ0 for all 0<ϵ<ϵ0 and σ∈SN−1,λ∈[12,1),

where δ0 is defined in Lemma 5.5. Now define ςϵ(λ, σ) : [0, 1] × 𝕊N–1 → 𝕊N–1 by

ςϵ(λ,σ)=y(2λ(s1−s2)+s2) if λ∈[0,12),G¯(gρ2(1−λ)ϵ) if λ∈[12,1),σ if λ=1.

Clearly, ςϵ is well defined. We claim that limλ→1− ςϵ(λ, σ) = σ and limλ→12− ςϵ(λ, σ) = G¯(gρϵ,σ) .

  1. limλ→1− ςϵ(λ, σ) = σ: Indeed

    ∫RNx|x||∇gρ2(1−λ)ϵ|2 dx=SH,L2N−μN−μ+2σ+o(1) as λ→1−

    then limλ→1− ςϵ(λ, σ) = σ.

  2. limλ→12− ςϵ(λ, σ) = G¯(gρϵ,σ) : Indeed

    limλ→12−ςϵ(λ,σ)=limλ→12−y(2λ(s1−s2)+s2)=y(s1)=G¯(gρϵ,σ).

Hence, ςϵ ∈ C([0, 1] × 𝕊N–1, 𝕊N–1) and ςϵ(0, σ) = G(Hϵ(σ)) and ςϵ(1, σ) = σ for σ ∈ 𝕊N–1 provided 0 < ϵ < ϵ0 and ∥f∥H–1 < e22. Thus the result follows.□

Proposition 5.9

Let e* := min{e00, e11, e22} where e00; e11 and e22 defined in Lemma 3.1, Lemma 5.4 and Lemma 5.7 respectively. Let ∥f∥H–1 < e* then 𝓙f has two critical points in

[Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2].

Equivalently, (Pf) have another two different solutions which are different from u1.

Proof

Using Lemma 5.8 and Lemma 5.3, we have

cat([Jf≤Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2−δϵ])≥2.

Now the proof follows from Lemma 4.8(i) and Lemma 5.2.□

6 Existence of Fourth solution

In this section we will prove the existence of high energy solution by using Brouwer’s degree theory and minmax theorem given by Brezis and Nirenberg [8].

Let V:={u∈H01(Ω):∫Ω∫Ω|u+(x)|2μ∗|u+(y)|2μ∗|x−y|μ dxdy=1},hρϵ,σ(x)=gρϵ,σ(x)∥gρϵ,σ∥NL where gρϵ,σ is defined in (4.1).

Lemma 6.1

∥hρϵ,σ∥D1,2(RN)2→SH,L as ϵ → 0 uniformly in σ ∈ 𝕊N–1.

Proof

Proof follows from Lemma 4.1(i).□

Lemma 6.2

There exists a ρ0 > 0 such that for 0 < ρ < ρ0, supσ∈SN−1,ϵ∈(0,1]∥hρϵ,σ∥2<2N−μ+22N−μSH,L.

Proof

Since we know that ∥∇uϵσ∥L2(RN)2=∥uϵσ∥NL2.2μ∗=SH,L2N−μN−μ+2 and this on using Lemma 4.2 we get supσ∈SN−1,ϵ∈(0,1]∥hρϵ,σ∥2→SH,L as ρ → 0. So there exists a ρ0 such that 0 < ρ < ρ0, we obtain supσ∈SN−1,ϵ∈(0,1]∥hρϵ,σ∥2<2N−μ+22N−μSH,L.

□

Now for any u ∈ H01 (Ω), by extending it to be zero outside Ω, we define Barycenter mapping β : 𝓥 → ℝN as

β(u)=∫RN∫RNx|u+(x)|2μ∗|u+(y)|2μ∗|x−y|μ dxdy,

and also let 𝓠 := {u ∈ 𝓥 : β(u) = 0}.

Lemma 6.3

There holds limϵ→0β(hρϵ,σ)=σ.

Proof

If there exists η > 0 and a sequence ϵn → 0+ such that |β(hρϵn)−σ| ≥ η. Then

β(hρϵn)=∫RN∫RNx|hρϵn(x)|2μ∗|hρϵn(y)|2μ∗|x−y|μ dxdy∥hρϵn∥NL2.2μ∗=σ+ϵn∫RN∫RN(z−σ)|υρ(ϵnz+(1−ϵn)σ)|2μ∗|υρ(ϵnw+(1−ϵn)σ)|2μ∗|z−w|μ[1+|z|2]2N−μ2[1+|w|2]2N−μ2 dzdw∫RN∫RN|υρ(ϵnz+(1−ϵn)σ)|2μ∗|υρ(ϵnw+(1−ϵn)σ)|2μ∗|z−w|μ[1+|z|2]2N−μ2[1+|w|2]2N−μ2 dzdw≤σ+ϵnsupz∈supp(υρ)|z−σ|≤σ+Cϵn, for some C>0.

It implies that 0 < η ≤ |β(hρϵn)−σ| ≤ Cϵn → 0+ as ϵn → 0+, a contradiction.□

Lemma 6.4

Let m0 = infu∈Q ∥u∥2 then SH,L < m0.

Proof

Obviously SH,L ≤ m0, so let if possible, SH,L = infu∈Q ∥u∥2 then there exists a sequence {vn} ∈ H01 (Ω) such that ∥vn∥NL = 1, β(vn) = 0, ∥vn∥2 → SH,L as n → ∞. Setting wn=SH,LN−22(N−μ+2)vn we get ∥wn∥NL2.2μ∗=SH,L2N−μN−μ+2 and ∥wn∥2→SH,L2N−μN−μ+2. Therefore, J(wn)→N−μ+22(2N−μ)SH,L2N−μN−μ+2 and 𝓙′(wn)(wn) = o(1). Using Lemma 2.9, we obtain {wn} is a Palais-Smale sequence of 𝓙 at level N−μ+22(2N−μ)SH,L2N−μN−μ+2. Subsequently, by [19, Theorem 4.4] and Remark 2.8, there exist sequences yn ∈ Ω, λn ∈ ℝ+ such that yn → y0 ∈ Ω and λn → 0, for the functions

vn=SH,L−N−22(N−μ+2)wn, where wn=Cλnλn2+|x−yn|2N−22 for some C>0.

Thus if

C1=C∫RN∫RNz|z−w|μ[1+|z|2]2N−μ2[1+|w|2]2N−μ2 dzdw and C2=C∫RN∫RN1|z−w|μ[1+|z|2]2N−μ2[1+|w|2]2N−μ2 dzdw,

then

0=β(vn)=C∫RN∫RNx|vn(x)|2μ∗|vn(y)|2μ∗|x−y|μ dxdy=λnC1+ynC2→C2y0.

This is a contradiction. Hence SH,L < m0.□

Lemma 6.5

There exists ϵ0 > 0 such that for 0 < ϵ < ϵ0 and |σ| = 1 we have

SH,L<∥hρϵ,σ∥D1,2(RN)2<m0+SH,L2.

Proof

Apparently SH,L ≤ ∥hρϵ,σ∥D1,2(RN)2 and we know that SH,L is not attained on a bounded domain. Thus, SH,L < ∥hρϵ,σ∥D1,2(RN)2 . Since SH,L < m0, there exists δ0 such that SH,L2+δ0<m02 and from Lemma 6.1 we know that ∥hρϵ,σ∥D1,2(RN)2 → SH,L as ϵ → 0. Therefore for δ0 > 0 there exists a ϵ0 > 0 such that ∥hρϵ,σ∥D1,2(RN)2 < SH,L + δ0 whenever 0 < ϵ < ϵ0. Hence we have the desired result.□

Now we will state the minimax lemma given by Brezis and Nirenberg [8].

Lemma 6.6

Let Y be a Banach space and ϕ ∈ C1(Y, ℝ). Let A be a compact metric space, A0 ⊂ A be a closed set and y ∈ C(A0, Y). Define

Γ={g∈C(A,Y):g(s)=y(s) if s∈A0},c¯=infg∈Γsups∈Aϕ(g(s)),c^=supy(A0)ϕ.

If c > ĉ then there exists a sequence {un} ∈ Y satisfying ϕ(un) → c and ϕ′(un) → 0.. Further, if ϕ satisfies (PS)c condition then there exists u0 ∈ Y such that ϕ(u0) = c and ϕ′(u0) = 0.

Let r0 = 1 – ϵ0 and Br0 = {(1 – ϵ)σ ∈ ℝN : |(1 – ϵ)σ| ≤ r0, σ ∈ 𝕊N–1, 0 < ϵ ≤ 1}, where ϵ0 is defined in Lemma 6.5. Then we set F={q∈C(B¯r0,V);q|∂B¯r0=hρϵ,σ} and

c¯=infq∈Fsup(1−ϵ)σ∈B¯r0∥q((1−ϵ)σ)∥2,c^=sup∂B¯r0∥hρϵ,σ∥2

Lemma 6.7

For each q ∈ F, we have q(Br0) ∩ 𝓠 ≠ ∅.

Proof

It is enough to show there exist ẽ > 0 and σ̃ ∈ 𝕊N–1 such that β(q((1 – ẽ)σ̃)) = 0. Define ψ : Br0 → ℝN by ψ((1 – ϵ)σ) = β(q((1 – ϵ)σ)). We claim that

d(ψ,B¯r0,0)=d(I,B¯r0,0)≠0, where d is Brouwer's topological degree.

If (1 – ϵ)σ ∈ ∂Br0 then q((1 – ϵ)σ) = hρϵ,σ which implies

ψ((1−ϵ)σ)=β(q((1−ϵ)σ))=β(hρϵ,σ)=σ+o(1) as ϵ→0.

Now define the homotopy 𝓗 : [0, 1] × Br0 → ℝN by

H(t,(1−ϵ)σ)=(1−t)ψ((1−ϵ)σ)+tI((1−ϵ)σ)

then for (1 – ϵ)σ ∈ ∂Br0 and t ∈ [0, 1] we have

H(t,(1−ϵ)σ)=(1−t)σ+o(1)+t(1−ϵ0)σ=o(1)+(1−ϵ0t)σ≠0, as ϵ→0.

So by Brouwer’s degree theory, claim holds. It implies that there exist ẽ > 0 and σ̃ ∈ 𝕊N–1 such that ψ((1 – ẽ)σ̃) = 0 that is, β(q((1 – ẽ)σ̃)) = 0.□

Using above Lemma we have m0≤sup(1−ϵ)σ∈B¯r0∥q((1−ϵ)σ)∥2 for all q ∈ F. Hence

m0≤infq∈Fsup(1−ϵ)σ∈B¯r0∥q((1−ϵ)σ)∥2=c¯.

Also, by the definition of c, and Lemma 6.2, we have c¯<2N−μ+22N−μSH,L for 0 < ρ < ρ0. Combining all these and using Lemma 6.4 we have

SH,L<m0≤c¯<2N−μ+22N−μSH,L for ρ sufficiently small . (6.1)

In addition, from Lemma 6.5, we get

c^=sup∂B¯r0∥hρϵ,σ∥2<m0+SH,L2<m0≤c¯.

Now we define

J˘f(u)=maxt>0Jf(tu):V→RNand J˘(u)=maxt>0J(tu):V→RN,yf=infq∈Fsup(1−ϵ)σ∈B¯r0J˘f(q((1−ϵ)σ))and y0=infq∈Fsup(1−ϵ)σ∈B¯r0J˘(q((1−ϵ)σ)).

We remark that the conclusion of Lemma 5.4 (iii) holds true for 𝓙̆f. Moreover, 𝓙̆f(u) = maxt>0 𝓙f(tu) = 𝓙f(t–(u)u), where t–(u) is defined in Lemma 3.3.

Lemma 6.8

The following holds:

  1. 𝓙̆f ∈ C1(𝓥, ℝ) and 〈J˘f′(u),h〉=t−(u)〈Jf′(t−(u)u),h〉 for all h ∈ Tu(𝓥) where Tu(V):={h∈H01(Ω)|∫Ω∫Ω|u+(x)|2μ∗|u+(y)|2μ∗−1h(y)|x−y|μ=0}.

  2. If u ∈ 𝓥 is a critical point of 𝓙̆f then t–(u)u ∈ Nf− is a critical point of 𝓙f.

  3. If {un}n∈ℕ is a (PS)c sequence of 𝓙̆f then {t–(un)un}n∈ℕ ∈ Nf− is a (PS)c sequence for 𝓙f.

Proof

  1. For every u ∈ H01 (Ω), t–(u)u ∈ Nf− that is, 〈 Jf′ (t–(u)u), u〉 = 0 and d2dt2|t=t−(u)Jf(tu)<0. Therefore, by implicit function theorem, we get t–(u) ∈ C1(𝓥, (0, ∞)). As a result, 𝓙̆f(u) = 𝓙f(t–(u)u) ∈ C1(𝓥, ℝ) and for all h ∈ Tu(𝓥), we have

    〈J˘f′(u),h〉=t−(u)〈Jf′(t−(u)u),h〉+〈Jf′(t−(u)u),u〉〈(t−(u))′,h〉=t−(u)〈Jf′(t−(u)u),h〉.

  2. Combining the fact that u ∈ 𝓥 is a critical point of 𝓙̆f and 〈 Jf′ (t–(u)u), u〉 = 0, we get the desired result.

  3. Let {un}n∈ℕ is a (PS)c sequence of 𝓙̆f, that is, un ∈ 𝓥, 𝓙̆f(un) → c and

    ∥J˘f′(u)∥Tun∗(V)=sup{|〈J˘f′(un),h〉|:h∈Tun(V),∥h∥=1}→0 as n→∞.

    By Lemma 3.3 we have t−(un)>∥u∥22.2μ∗−112.2μ∗−2>SH,L2.2μ∗−112.2μ∗−2>C for some C > 0. Since H01 (Ω) = Run ⊕ Tun(𝓥) so 〈 Jf′ (un), v〉 = 〈 Jf′ (un), hv 〉, where hv is the projection of v in Tun(𝓥). Hence,

    ∥Jf′(t−(un)un)∥=supv∈H01(Ω),∥v∥=1|〈Jf′(t−(un)un),v〉|=supv∈H01(Ω),∥v∥=1|〈Jf′(t−(un)un),hv〉|=supv∈H01(Ω),∥v∥=11t−(un)|〈J˘f′(un),hv〉|≤1C∥J˘f′(u)∥Tun∗(V)→0.

    Clearly, 𝓙f(t–(un)un) → c. Therefore, {t–(un)un}n∈ℕ ∈ Nf− is a (PS)c sequence for 𝓙f.□

Lemma 6.9

If 0 < ρ < ρ0, then N−μ+22(2N−μ)SH,L2N−μN−μ+2<y0<N−μ+22N−μSH,L2N−μN−μ+2.

Proof

For u ∈ 𝓥, solving J′(tu)=ta(u)−t2.2μ∗−1=0 we get t = 0 and t=(a(u))12.2μ∗−2. Therefore,

J˘(u)=maxt>0J(tu)=N−μ+22(2N−μ)∥u∥2(2N−μ)N−μ+2.

From the definition of c, we obtain

y0=N−μ+22(2N−μ)infq∈Fsup(1−ϵ)σ∈B¯r0∥q((1−ϵ)σ)∥2(2N−μ)N−μ+2=N−μ+22(2N−μ)c¯2N−μN−μ+2

which on using (6.1) yields the desired result.□

Lemma 6.10

J˘f(hρϵ,σ)=N−μ+22(2N−μ)SH,L2N−μN−μ+2+o(1) as ϵ → 0.

Proof

By Lemma 4.1, hρϵ,σ ⇀ 0 in H01 (Ω) as ϵ → 0. On solving

Jf′(thρϵ,σ)=t a(hρϵ,σ)−t2.2μ∗−1−∫Ωfhρϵ,σ dx=0,

we conclude tf=∥gρϵ,σ∥NL+o(1). Hence again from the Lemma 4.1 we obtain

J˘f(hρϵ,σ)=maxt>0Jf(thρϵ,σ)=Jf(tfhρϵ,σ)=Jf(gρϵ,σ)=N−μ+22(2N−μ)SH,L2N−μN−μ+2+o(1) as ϵ→0.

□

Lemma 6.11

There exists e0∗ > 0 such that if 0 < ∥f∥H−1 < e0∗ ,

Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2<yf<Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

Proof

Analogous to the proof of Lemma 5.4(iii) we can have

(1−ω)2N−μN−μ+2J(t∗u)−12ω∥f∥H−12≤Jf(t−u)≤(1+ω)2N−μN−μ+2J(t∗u)+12ω∥f∥H−12.

Using the above inequality with the definition of 𝓙̆ and 𝓙̆f, we get

(1−ω)2N−μN−μ+2J(t∗u)−12ω∥f∥H−12≤Jf(t−u)≤(1+ω)2N−μN−μ+2J(t∗u)+12ω∥f∥H−12.

For δ > 0 there exists e1(δ) such that if ∥f∥H−1 < e1(δ) then

y0−δ<yf<y0+δ. (6.2)

Now from Lemma 5.4(iii) for each 0 < ω < 1, we have

(1−ω)2N−μN−μ+2N−μ+22(2N−μ)SH,L2N−μN−μ+2−12ω∥f∥H−12≤Υf−(Ω)≤(1+ω)2N−μN−μ+2N−μ+22(2N−μ)SH,L2N−μN−μ+2+12ω∥f∥H−12.

So for δ > 0 there exists e2(δ) > 0 such that whenever ∥f∥H−1 < e2(δ) then

N−μ+22(2N−μ)SH,L2N−μN−μ+2−δ≤Υf−(Ω)≤N−μ+22(2N−μ)SH,L2N−μN−μ+2+δ.

It implies

N−μ+22N−μSH,L2N−μN−μ+2−δ≤Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2≤N−μ+22N−μSH,L2N−μN−μ+2+δ. (6.3)

Moreover, from Lemma 6.9

N−μ+22(2N−μ)SH,L2N−μN−μ+2<y0<N−μ+22N−μSH,L2N−μN−μ+2.

Hence for fix small 0 < ϵ < min{N−μ+22N−μSH,L2N−μN−μ+2−y02,y0−N−μ+22(2N−μ)SH,L2N−μN−μ+2} such that if ∥f∥H−1 < e0∗ = min{e2(ϵ), e2(ϵ)} then using (6.2) and (6.3), we obtain

Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2<N−μ+22(2N−μ)SH,L2N−μN−μ+2<y0−ϵ≤yf and yf<y0+2ϵ−ϵ<N−μ+22N−μSH,L2N−μN−μ+2−ϵ≤Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2.

That is, Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2<yf<Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2. □

Proposition 6.12

If 0 < ρ < ρ0, 0 < ∥f∥H−1 < e0∗ (defined in Lemma 6.11) then there exists a critical point u4 ∈ Nf− of 𝓙f with 𝓙f(u4) = yf.

Proof

Let c∈Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2,Υf−(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2 and {un}n∈ℕ is a (PS)c sequence of 𝓙̆f. Then by Lemma 6.8, {t−(un)un}n∈ℕ ∈ Nf− is a (PS)c sequence for 𝓙f which on using Lemma 4.8 gives that {un}n∈ℕ is compact. Moreover, from Lemma 6.10, yf>J˘f(hρϵ,σ)=N−μ+22(2N−μ)SH,L2N−μN−μ+2+o(1) as ϵ sufficiently small. Using Lemma 6.6 we have yf is a critical value of 𝓙̆f. Therefore, there exists v4 ∈ 𝓥 such that 𝓙̆f(v4) = yf and J˘f′(v4) = 0. Thus by Lemma 6.8, u4 := t−(v4) v4 ∈ Nf− is a critical point of 𝓙f and 𝓙f(u4) = yf.□

Proof of Theorem 1.1

First note that by Lemma 3.8, we have all solutions of (Pf) are positive in Ω and from Lemma 3.7, we have u1 ∈ Nf+ ⊂ H01 (Ω) such that 𝓙f(u1) = Υf whenever 0 < ∥f∥H−1 < e00. By Proposition 5.9 we have two more critical point u2, u3 ∈ Nf− of 𝓙f such that in 𝓙f(u2), 𝓙f(u3) < Υf(Ω) + N−μ+22(2N−μ)SH,L2N−μN−μ+2. Therefore we get three positive solutions of (Pf) whenever 0 < ∥f∥H−1 < e* where e* is defined in Proposition 5.9. Let e** = min{e*, e0∗ } then by Proposition 6.12, we get u4 ∈ Nf− 𝓙f(u4) = yf.

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Received: 2018-12-18
Accepted: 2019-03-08
Published Online: 2019-08-06

© 2020 D. Goel and K. Sreenadh, published by De Gruyter

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

Artikel in diesem Heft

  1. Frontmatter
  2. On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
  3. Regularity of solutions of the parabolic normalized p-Laplace equation
  4. Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
  5. Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
  6. Radon measure-valued solutions of first order scalar conservation laws
  7. Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
  8. Generalized solutions of variational problems and applications
  9. Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
  10. Nonlinear Sherman-type inequalities
  11. Global regularity for systems with p-structure depending on the symmetric gradient
  12. Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
  13. Noncoercive resonant (p,2)-equations with concave terms
  14. Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
  15. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
  16. Localization and multiplicity in the homogenization of nonlinear problems
  17. Remarks on a nonlinear nonlocal operator in Orlicz spaces
  18. A Picone identity for variable exponent operators and applications
  19. On the weakly degenerate Allen-Cahn equation
  20. Continuity results for parametric nonlinear singular Dirichlet problems
  21. Construction of type I blowup solutions for a higher order semilinear parabolic equation
  22. Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
  23. Comparison results for nonlinear divergence structure elliptic PDE’s
  24. Constant sign and nodal solutions for parametric (p, 2)-equations
  25. Monotonicity formulas for coupled elliptic gradient systems with applications
  26. Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
  27. A class of semipositone p-Laplacian problems with a critical growth reaction term
  28. The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
  29. Reconstruction of Tesla micro-valve using topological sensitivity analysis
  30. Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
  31. Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
  32. Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
  33. Homoclinics for singular strong force Lagrangian systems
  34. A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
  35. On a class of nonlocal nonlinear Schrödinger equations with potential well
  36. Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
  37. Regularity for minimizers for functionals of double phase with variable exponents
  38. Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
  39. Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
  40. A-priori bounds for quasilinear problems in critical dimension
  41. Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
  42. On the Sobolev space of functions with derivative of logarithmic order
  43. On a logarithmic Hartree equation
  44. Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
  45. Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
  46. Existence for (p, q) critical systems in the Heisenberg group
  47. Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
  48. Some hemivariational inequalities in the Euclidean space
  49. Existence of standing waves for quasi-linear Schrödinger equations on Tn
  50. Periodic solutions for second order differential equations with indefinite singularities
  51. On the Hölder continuity for a class of vectorial problems
  52. Bifurcations of nontrivial solutions of a cubic Helmholtz system
  53. On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
  54. Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
  55. Positive solutions for diffusive Logistic equation with refuge
  56. Null controllability for a degenerate population model in divergence form via Carleman estimates
  57. Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
  58. On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
  59. Multiplicity and concentration results for magnetic relativistic Schrödinger equations
  60. Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
  61. The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
  62. Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
  63. Pseudo almost periodic solutions for a class of differential equation with delays depending on state
  64. Normalized multi-bump solutions for saturable Schrödinger equations
  65. Some inequalities and superposition operator in the space of regulated functions
  66. Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
  67. Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
  68. Morrey estimates for a class of elliptic equations with drift term
  69. A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
  70. Global and non global solutions for a class of coupled parabolic systems
  71. On the analysis of a geometrically selective turbulence model
  72. Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
  73. Lack of smoothing for bounded solutions of a semilinear parabolic equation
  74. Gradient estimates for the fundamental solution of Lévy type operator
  75. π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
  76. On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
  77. Anisotropic problems with unbalanced growth
  78. On a fractional thin film equation
  79. Minimum action solutions of nonhomogeneous Schrödinger equations
  80. Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
  81. Optimal rearrangement problem and normalized obstacle problem in the fractional setting
  82. A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
https://doi.org/10.1515/anona-2020-0026
Schlagwörter für diesen Artikel
Hardy-Littlewood-Sobolev inequality; critical problems; non-contractible domains
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Frontmatter
  2. On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
  3. Regularity of solutions of the parabolic normalized p-Laplace equation
  4. Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
  5. Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
  6. Radon measure-valued solutions of first order scalar conservation laws
  7. Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
  8. Generalized solutions of variational problems and applications
  9. Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
  10. Nonlinear Sherman-type inequalities
  11. Global regularity for systems with p-structure depending on the symmetric gradient
  12. Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
  13. Noncoercive resonant (p,2)-equations with concave terms
  14. Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
  15. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
  16. Localization and multiplicity in the homogenization of nonlinear problems
  17. Remarks on a nonlinear nonlocal operator in Orlicz spaces
  18. A Picone identity for variable exponent operators and applications
  19. On the weakly degenerate Allen-Cahn equation
  20. Continuity results for parametric nonlinear singular Dirichlet problems
  21. Construction of type I blowup solutions for a higher order semilinear parabolic equation
  22. Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
  23. Comparison results for nonlinear divergence structure elliptic PDE’s
  24. Constant sign and nodal solutions for parametric (p, 2)-equations
  25. Monotonicity formulas for coupled elliptic gradient systems with applications
  26. Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
  27. A class of semipositone p-Laplacian problems with a critical growth reaction term
  28. The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
  29. Reconstruction of Tesla micro-valve using topological sensitivity analysis
  30. Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
  31. Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
  32. Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
  33. Homoclinics for singular strong force Lagrangian systems
  34. A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
  35. On a class of nonlocal nonlinear Schrödinger equations with potential well
  36. Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
  37. Regularity for minimizers for functionals of double phase with variable exponents
  38. Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
  39. Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
  40. A-priori bounds for quasilinear problems in critical dimension
  41. Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
  42. On the Sobolev space of functions with derivative of logarithmic order
  43. On a logarithmic Hartree equation
  44. Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
  45. Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
  46. Existence for (p, q) critical systems in the Heisenberg group
  47. Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
  48. Some hemivariational inequalities in the Euclidean space
  49. Existence of standing waves for quasi-linear Schrödinger equations on Tn
  50. Periodic solutions for second order differential equations with indefinite singularities
  51. On the Hölder continuity for a class of vectorial problems
  52. Bifurcations of nontrivial solutions of a cubic Helmholtz system
  53. On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
  54. Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
  55. Positive solutions for diffusive Logistic equation with refuge
  56. Null controllability for a degenerate population model in divergence form via Carleman estimates
  57. Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
  58. On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
  59. Multiplicity and concentration results for magnetic relativistic Schrödinger equations
  60. Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
  61. The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
  62. Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
  63. Pseudo almost periodic solutions for a class of differential equation with delays depending on state
  64. Normalized multi-bump solutions for saturable Schrödinger equations
  65. Some inequalities and superposition operator in the space of regulated functions
  66. Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
  67. Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
  68. Morrey estimates for a class of elliptic equations with drift term
  69. A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
  70. Global and non global solutions for a class of coupled parabolic systems
  71. On the analysis of a geometrically selective turbulence model
  72. Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
  73. Lack of smoothing for bounded solutions of a semilinear parabolic equation
  74. Gradient estimates for the fundamental solution of Lévy type operator
  75. π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
  76. On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
  77. Anisotropic problems with unbalanced growth
  78. On a fractional thin film equation
  79. Minimum action solutions of nonhomogeneous Schrödinger equations
  80. Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
  81. Optimal rearrangement problem and normalized obstacle problem in the fractional setting
  82. A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
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