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Volume 9, Issue 1

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

Divya Goel
• Corresponding author
• Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-110016, India
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Published Online: 2019-08-06 | DOI: https://doi.org/10.1515/anona-2020-0026

## 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), $\begin{array}{}{2}_{\mu }^{\ast }=\frac{2N-\mu }{N-2}\end{array}$, fL(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.

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 $\begin{array}{}{2}^{\ast }=\frac{2N}{N-2}\end{array}$ is the critical Sobolev exponent and Ω is a bounded domain in ℝN with smooth boundary. If fH−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|

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 $\begin{array}{}{2}_{\mu }^{\ast }=\frac{2N-\mu }{N-2}\end{array}$, is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality (2.1) and f with := {f : fL(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 $\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ (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}, fL(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 < ∥fH−1 < e*. Moreover, if R1 is small enough then there exists e** > 0 such that (Pf) has at least four positive solutions whenever 0 < ∥fH−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, fLt(ℝN) and hLr(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/(2Nμ), 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 $\begin{array}{}{2}^{\ast }=\frac{2N}{N-2}\end{array}$)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 : $\begin{array}{}{H}_{0}^{1}\end{array}$ (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 𝓙fC1($\begin{array}{}{H}_{0}^{1}\end{array}$ (Ω), ℝ) and moreover, if μ < min {4, N} then 𝓙fC2($\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), ℝ).

#### Definition 2.4

A function u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) is called a weak solution of the problem (Pf) if for all v$\begin{array}{}{H}_{0}^{1}\end{array}$(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 $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) for 𝓙f if 𝓙f = c + o(1) and $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$ (un) = o(1) strongly in H−1 as n → ∞. We say 𝓙f satisfies the (PS)c condition in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) if every (PS)c sequence in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) has a convergent subsequence.

Since 𝓙f is not bounded below on $\begin{array}{}{H}_{0}^{1}\end{array}$(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 $\begin{array}{}{\mathit{Υ}}_{0}\left(\mathit{\Omega }\right):={\mathit{Υ}}_{0}=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$

Notations: Throughout the paper we will use the notation 𝓙0 = 𝓙, 𝓝0 = 𝓝, ∥.∥ = $\begin{array}{}\parallel .{\parallel }_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\end{array}$

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

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

#### Proposition 2.6

For any u, v$\begin{array}{}{H}_{0}^{1}\end{array}$(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$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), there exists a unique t > 0 such that t u ∈ 𝓝. Moreover, there holds $\begin{array}{}{\mathit{Υ}}_{0}\le \left(\frac{N-\mu +2}{2\left(2N-\mu \right)}\right){\left(\frac{a\left(u{\right)}^{{2}_{\mu }^{\ast }}}{b\left(u\right)}\right)}^{\frac{1}{{2}_{\mu }^{\ast }-1}}.\end{array}$

#### Proof

Let $\begin{array}{}{m}_{u}\left(t\right)=\frac{{t}^{2}}{2}a\left(u\right)-\frac{{t}^{{2.2}_{\mu }^{\ast }}}{{2.2}_{\mu }^{\ast }}b\left(u\right)\end{array}$ then on solving $\begin{array}{}{m}_{u}^{\mathrm{\prime }}\left(t\right)\end{array}$ = 0, we get unique $\begin{array}{}t\left(u\right)={\left(\frac{a\left(u\right)}{b\left(u\right)}\right)}^{\frac{1}{2\left({2}_{\mu }^{\ast }-1\right)}}\end{array}$ 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 $\begin{array}{}\mathcal{J}\left(u\right)>{\mathit{Υ}}_{0}=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$

#### Lemma 2.9

A sequence {un} is a (PS)Υ0- sequence for 𝓙 in $\begin{array}{}{H}_{0}^{1}\end{array}$(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) = $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}b\left({u}_{n}\right)\end{array}$ + on(1) and hence we have

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

Define $\begin{array}{}{T}_{n}\left(\psi \right)=\underset{\mathit{\Omega }}{\int }\underset{\mathit{\Omega }}{\int }\frac{\left({u}_{n}^{+}\left(x\right){\right)}^{{2}_{\mu }^{\ast }}\left({u}_{n}^{+}\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}\psi \left(y\right)}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\end{array}$ for ψ$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and n = 1, 2, ….

Claim: $\begin{array}{}\parallel {T}_{n}{\parallel }_{{H}^{-1}}=\left(D{\mathit{Υ}}_{0}{\right)}^{\frac{1}{2}}+{o}_{n}\left(1\right).\end{array}$

Let ψ$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that ∥ψ∥ = 1 then by Lemma 2.7, we know that there exists a t > 0 such that a() = b(). Therefore, $\begin{array}{}t=\parallel \psi {\parallel }_{NL}^{-\frac{{2}_{\mu }^{\ast }}{{2}_{\mu }^{\ast }-1}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathit{Υ}}_{0}\le \frac{1}{D}\parallel \psi {\parallel }_{NL}^{-\frac{{2.2}_{\mu }^{\ast }}{{2}_{\mu }^{\ast }-1}}.\end{array}$ 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 $\begin{array}{}\parallel {T}_{n}{\parallel }_{{H}^{-1}}\le \left(D{\mathit{Υ}}_{0}{\right)}^{\frac{1}{2}}+{o}_{n}\left(1\right).\end{array}$ Moreover, $\begin{array}{}{T}_{n}\left(\frac{{u}_{n}}{\parallel {u}_{n}\parallel }\right)=\left(b\left({u}_{n}\right){\right)}^{\frac{1}{2}}=\left(D{\mathit{Υ}}_{0}{\right)}^{\frac{1}{2}}+{o}_{n}\left(1\right).\end{array}$ This implies ∥TnH−1 = $\begin{array}{}\left(D{\mathit{Υ}}_{0}{\right)}^{\frac{1}{2}}+{o}_{n}\left(1\right).\end{array}$ Hence the proof of claim follows. Now, by Riesz representation theorem, for each n, there exists vn$\begin{array}{}{H}_{0}^{1}\end{array}$(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) = 0 + 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$\begin{array}{}{H}_{0}^{1}\end{array}$(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 ≤ ∥unvn∥ = 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 $\begin{array}{}{\varphi }_{u}^{\mathrm{\prime }}\end{array}$ (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 $\begin{array}{}{\mathcal{N}}_{f}^{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathcal{N}}_{f}^{-}\end{array}$ 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 $\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$.

#### Lemma 3.1

If fF̂ andfH−1 < e00 := $\begin{array}{}{C}_{N,\mu }{S}_{H,L}^{\frac{{2}_{\mu }^{\ast }}{{2.2}_{\mu }^{\ast }-2}}\end{array}$ where $\begin{array}{}{C}_{N,\mu }=\left(\frac{1}{{2.2}_{\mu }^{\ast }-1}\right){\phantom{\rule{thinmathspace}{0ex}}}^{\frac{{2.2}_{\mu }^{\ast }-1}{{2.2}_{\mu }^{\ast }-2}}\left({2.2}_{\mu }^{\ast }-2\right)\end{array}$ then $\begin{array}{}{\alpha }_{0}:=\underset{u\in E}{inf}\left\{{C}_{N,\mu }\parallel u{\parallel }^{\frac{{2.2}_{\mu }^{\ast }-1}{{2}_{\mu }^{\ast }-1}}-\underset{\mathit{\Omega }}{\int }fu\text{\hspace{0.17em}}dx\right\}\end{array}$ 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 $\begin{array}{}a\left(u\right)-\left({2.2}_{\mu }^{\ast }-1\right)b\left(u\right)\ne 0.\end{array}$ In particular, $\begin{array}{}{\mathcal{N}}_{f}^{0}\end{array}$ = {0}.

#### Lemma 3.3

For each u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) with u+ ≢ 0 the following holds:

1. There exists a unique t = t(u) > 0 such that t u $\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$(Ω). In particular,

$t−>a(u)(2.2μ∗−1)b(u)12.2μ∗−2:=tmax$

and $\begin{array}{}{\mathcal{J}}_{f}\left({t}^{-}u\right)=\underset{t\ge {t}_{max}}{max}{\mathcal{J}}_{f}\left(tu\right).\end{array}$

2. If $\begin{array}{}\underset{\mathit{\Omega }}{\int }fu>0\end{array}$, then there exists unique t+ ∈ (0, tmax) such that t+ u$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$ (Ω) and

$Jf(t+u)=min0

3. t(u) is a continuous function.

4. $\begin{array}{}{\mathcal{N}}_{f}^{-}=\left\{u\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\setminus \left\{0\right\}\phantom{\rule{thickmathspace}{0ex}}|\phantom{\rule{thickmathspace}{0ex}}{u}^{+}\not\equiv 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)=1\right\}.\end{array}$

#### Lemma 3.4

For each u$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$ (Ω), we haveΩ fu dx > 0 and 𝓙f(u) < 0. In particular, $\begin{array}{}{\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)\le {\mathit{Υ}}_{f}^{+}\left(\mathit{\Omega }\right)<0.\end{array}$

#### Lemma 3.5

Let u ∈ 𝓝f(Ω) be such that $\begin{array}{}{\mathcal{J}}_{f}\left(u\right)=\underset{w\in {\mathcal{N}}_{f}\left(\mathit{\Omega }\right)}{min}{\mathcal{J}}_{f}\left(w\right)\end{array}$ = Υf(Ω) thenΩ fu dx > 0 and u is a solution of (Pf).

#### Lemma 3.6

𝓙f has Palais-Smale sequences at each of the levels $\begin{array}{}{\mathit{Υ}}_{f}\left(\mathit{\Omega }\right),\text{\hspace{0.17em}}{\mathit{Υ}}_{f}^{+}\left(\mathit{\Omega }\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathit{Υ}}_{f}^{-}\left(\mathit{\Omega }\right).\end{array}$

#### 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$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu1 strongly in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω). Moreover, u1 ∈ 𝓝f and is a solution to (Pf).

#### Proof

𝓙f is bounded below and coercive implies {un} is bounded in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω). So, there exists a subsequence still denoted by {un} such that unu1 weakly in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω). By [19, Lemma 4.2], we have $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(u1) = 0. In particular, u1 ∈ 𝓝f and $\begin{array}{}{\mathcal{J}}_{f}\left({u}_{1}\right)=\left(\frac{1}{2}-\frac{1}{2.2{\ast }_{\mu }}\right)a\left({u}_{1}\right)-\left(1-\frac{1}{2.2{\ast }_{\mu }}\right)\underset{\mathit{\Omega }}{\int }f{u}_{1}\text{\hspace{0.17em}}dx.\end{array}$ 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 = unu1 then by [19, Lemma 4.1], [15, Lemma 2.2] and the fact that $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$ (u1) = 0, we obtain 𝓙f(wn) = 𝓙f(un) − 𝓙f(u1) = on(1) and $\begin{array}{}〈{\mathcal{J}}_{f}^{\mathrm{\prime }}\left({w}_{n}\right),\varphi 〉=〈{\mathcal{J}}_{f}^{\mathrm{\prime }}\left({u}_{n}\right),\varphi 〉-〈{\mathcal{J}}_{f}^{\mathrm{\prime }}\left({u}_{1}\right),\varphi 〉+{o}_{n}\left(1\right)\end{array}$ = on(1). Therefore, 〈$\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(wn), wn〉 = on(1). It implies $\begin{array}{}{\mathcal{J}}_{f}\left({w}_{n}\right)=\left(\frac{1}{2}-\frac{1}{2.2{\ast }_{\mu }}\right)a\left({w}_{n}\right)-{\int }_{\mathit{\Omega }}f{w}_{n}\text{\hspace{0.17em}}dx={o}_{n}\left(1\right)\end{array}$ and since ∫Ω fwn dx = on(1), we get a(wn) = on(1). Hence unu strongly in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω).□

#### Lemma 3.8

If u be a solution of (Pf) then uC2(Ω). Moreover, u is a positive solution.

#### Proof

Let u be a solution of (Pf) and G(x, u) = $\begin{array}{}\left(\underset{\mathit{\Omega }}{\int }\frac{|{u}^{+}\left(y\right){|}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|{u}^{+}{|}^{{2}_{\mu }^{\ast }-2}u+f\end{array}$. By using same assertions and arguments as in [25, Proposition 3.1 and Theorem 2], we have $\begin{array}{}\left(\underset{\mathit{\Omega }}{\int }\frac{|{u}^{+}\left(y\right){|}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)\end{array}$L (Ω) and since f, we have |G(x, u)| ≤ C(1 + |u|2*−1). Then by the standard elliptic regularity uC2(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 $\begin{array}{}{k}_{0}={\left(\frac{1}{{2.2}_{\mu }^{\ast }-1}\right)}^{\frac{1}{2\left({2}_{\mu }^{\ast }-1\right)}}{S}_{H,L}^{\frac{{2}_{\mu }^{\ast }}{2\left({2}_{\mu }^{\ast }-1\right)}}\end{array}$ and f, ∥fH−1e00 (where e00 is defined in Lemma 3.1) then

1. $\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$(Ω) ⊂ Bk0(0) := {u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) | ∥u∥ < k0}.

2. 𝓙f is strictly convex in Bk0(0).

#### Proof

1. Let u$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$(Ω) then $\begin{array}{}{\varphi }_{u}^{\mathrm{\prime }}\left(1\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\varphi }_{u}^{\mathrm{\prime }\mathrm{\prime }}\left(1\right)>0.\end{array}$ That is, a(u) = b(u) + ∫Ω fu dx and a(u) > $\begin{array}{}\left({2.2}_{\mu }^{\ast }-1\right)b\left(u\right).\end{array}$ Therefore, a(u) = b(u) + ∫Ω fu dx < $\begin{array}{}\frac{1}{\left({2.2}_{\mu }^{\ast }-1\right)}a\left(u\right)\end{array}$ + ∫Ω fu dx. It implies $\begin{array}{}\left(1-\frac{1}{\left({2.2}_{\mu }^{\ast }-1\right)}\right)a\left(u\right)\le \parallel f{\parallel }_{{H}^{-1}}\parallel u\parallel .\end{array}$ 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 $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$(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 uBk0(0) ∖ {0}. Then $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$ (u) is positive definite for uBk0(0) and 𝓙f(u) is strictly convex on Bk0(0).□

#### Lemma 3.10

It holds that u1$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$ and 𝓙f(u1) = $\begin{array}{}{\mathit{Υ}}_{f}^{+}\end{array}$(Ω) = Υf(Ω). Moreover, u1 is the unique critical point of 𝓙f in Bk0(0) and u1 is a local minimum of 𝓙f in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω).

#### Proof

Using the proof of [30, Theorem 1.3], we have $\begin{array}{}\underset{\mathit{\Omega }}{\int }f{u}_{1}\text{\hspace{0.17em}}dx>0.\end{array}$ Now if u1$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ then there exists a unique t(u1) = 1 > tmax > t+(u1) > 0 such that t+(u1)u1$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$ then by Lemma 3.3 (b) we have

$Υf(Ω)≤Υf+(Ω)≤Jf(t+(u1)u1)

which is a contradiction. It implies u1$\begin{array}{}{\mathcal{N}}_{f}^{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathit{Υ}}_{f}^{+}\left(\mathit{\Omega }\right)\le {\mathcal{J}}_{f}\left({u}_{1}\right)={\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)\le {\mathit{Υ}}_{f}^{+}\left(\mathit{\Omega }\right)\end{array}$ that is, 𝓙f(u1) = Υf(Ω) = $\begin{array}{}{\mathit{Υ}}_{f}^{+}\end{array}$(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$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) be a critical point of 𝓙f then either u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ or u = u1.

#### Proof

If u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) be a critical point of 𝓙f then u ∈ 𝓝f = $\begin{array}{}{\mathcal{N}}_{f}^{+}\cup {\mathcal{N}}_{f}^{-}\end{array}$. Now using the fact that $\begin{array}{}{\mathcal{N}}_{f}^{+}\cup {\mathcal{N}}_{f}^{-}\end{array}$ = ∅ and $\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$Bk0(0) we have either u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ 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 $\begin{array}{}-\mathit{\Delta }u=\left(|x{|}^{-\mu }\ast |u{|}^{{2}_{\mu }^{\ast }}\right)|u{|}^{{2}_{\mu }^{\ast }-1}\end{array}$ 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, $\begin{array}{}\frac{1}{2}\end{array}$). Now, define υρ$\begin{array}{}{C}_{c}^{\mathrm{\infty }}\end{array}$(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. $\begin{array}{}a\left({g}_{\rho }^{ϵ,\sigma }\right)=b\left({g}_{\rho }^{ϵ,\sigma }\right)={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+{o}_{ϵ}\left(1\right)\end{array}$ uniformly in σ as ϵ → 0.

2. $\begin{array}{}\mathcal{J}\left({g}_{\rho }^{ϵ,\sigma }\right)=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+{o}_{ϵ}\left(1\right)\end{array}$ uniformly in σ as ϵ → 0.

3. $\begin{array}{}{g}_{\rho }^{ϵ,\sigma }\end{array}$ ⇀ 0 weakly in $\begin{array}{}{H}_{0}^{1}\end{array}$(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|(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, $\begin{array}{}\parallel \mathrm{\nabla }{g}_{\rho }^{ϵ,\sigma }{\parallel }_{{L}^{2}\left({\mathbb{R}}^{N}\right)}=\parallel \mathrm{\nabla }{u}_{ϵ}^{\sigma }{\parallel }_{{L}^{2}\left({\mathbb{R}}^{N}\right)}+{o}_{ϵ}\left(1\right)={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+{o}_{ϵ}\left(1\right).\end{array}$

Next we will prove that $\begin{array}{}b\left({g}_{\rho }^{ϵ,\sigma }\right)={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+{o}_{ϵ}\left(1\right)\end{array}$ 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 $\begin{array}{}{\xi }_{ϵ}\left(x\right)=\frac{{ϵ}^{N}}{\left({ϵ}^{2}+|x-\left(1-ϵ\right)\sigma {|}^{2}{\right)}^{N}}\end{array}$ then taking into account the definition of $\begin{array}{}{u}_{ϵ}^{\sigma }\end{array}$, (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, $\begin{array}{}b\left({g}_{\rho }^{ϵ,\sigma }\right)-\underset{{\mathbb{R}}^{N}}{\int }\underset{{\mathbb{R}}^{N}}{\int }\frac{|{u}_{ϵ}^{\sigma }\left(x\right){|}^{{2}_{\mu }^{\ast }}|{u}_{ϵ}^{\sigma }\left(y\right){|}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\to 0\end{array}$ as ϵ → 0 that is, $\begin{array}{}b\left({g}_{\rho }^{ϵ,\sigma }\right)\to {S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ as ϵ → 0 and completes the proof of (i).

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

3. Assume by contradiction, $\begin{array}{}{g}_{\rho }^{ϵ,\sigma }\end{array}$g1 ≢ 0 weakly in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) then $\begin{array}{}{g}_{\rho }^{ϵ,\sigma }\end{array}$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. $\begin{array}{}\underset{\rho \to 0}{lim}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\parallel \mathrm{\nabla }\left({g}_{\rho }^{ϵ,\sigma }-{u}_{ϵ}^{\sigma }\right){\parallel }_{{L}^{2}\left({\mathbb{R}}^{N}\right)}^{2}=0.\end{array}$

2. $\begin{array}{}\underset{\rho \to 0}{lim}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\parallel {g}_{\rho }^{ϵ,\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}=\parallel {u}_{ϵ}^{\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}.\end{array}$

#### 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 $\begin{array}{}{u}_{ϵ}^{\sigma }\end{array}$, 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 $\begin{array}{}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\left(\parallel {g}_{\rho }^{ϵ,\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}-\parallel {u}_{ϵ}^{\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}\right)\to 0\end{array}$ as ρ → 0 and completes the proof.□

#### Lemma 4.3

The following asymptic estimates hold:

1. $\begin{array}{}a\left({g}_{\rho }^{ϵ,\sigma }\right)\le {S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+O\left({ϵ}^{N-2}\right).\end{array}$

2. $\begin{array}{}b\left({g}_{\rho }^{ϵ,\sigma }\right)\le {S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+O\left({ϵ}^{N}\right).\end{array}$

3. $\begin{array}{}b\left({g}_{\rho }^{ϵ,\sigma }\right)\ge {S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}-O\left({ϵ}^{\frac{2N-\mu }{2}}\right).\end{array}$

#### Proof

Part (i) follows from Lemma 4.1 (i). For part (ii) we will first estimate the integral $\begin{array}{}\underset{\mathit{\Omega }}{\int }|{g}_{\rho }^{ϵ,\sigma }{|}^{{2}^{\ast }}\text{\hspace{0.17em}}dx.\end{array}$ 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 $\begin{array}{}\underset{\mathit{\Omega }}{\int }|{g}_{\rho }^{ϵ,\sigma }{|}^{{2}^{\ast }}\text{\hspace{0.17em}}dx\le {S}^{\frac{N}{N-\mu +2}}C\left(N,\mu {\right)}^{\frac{-N}{N-\mu +2}}+O\left({ϵ}^{N}\right)\end{array}$ 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 $\begin{array}{}\parallel {u}_{ϵ}^{\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ + 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:

• Case 1

$\begin{array}{}{2}_{\mu }^{\ast }\end{array}$ > 3.

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

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

which implies that

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

• Case 2

2 < $\begin{array}{}{2}_{\mu }^{\ast }\end{array}$ ≤ 3.

We recall the inequality from [7, Lemma 4]: there exist C(depending on $\begin{array}{}{2}_{\mu }^{\ast }\end{array}$) 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 = O1O2O3O4, 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)

Also, define the $\begin{array}{}b\left(u{\right)}_{|{O}_{i}}=\int \underset{{O}_{i}}{\int }\frac{\left(u\left(x\right){\right)}^{{2}_{\mu }^{\ast }}\left(u\left(y\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\end{array}$, for all u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and i = 1, 2, 3, 4.

• Subcase 1

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 $\begin{array}{}{A}_{ϵ}^{1}\end{array}$ is sum of eight non-negative integrals and each integral has an upper bound of the form $\begin{array}{}C\int {\int }_{{O}_{1}}\frac{{u}_{1}\left(x\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{{2}_{\mu }^{\ast }-1}\left({u}_{1}\left(y\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}C\int {\int }_{{O}_{1}}\frac{{u}_{1}\left(y\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}\left({u}_{1}\left(x\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy.\end{array}$ Write $\begin{array}{}\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{{2}_{\mu }^{\ast }-1}\end{array}$ = $\begin{array}{}\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{r}.\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{s}\end{array}$ with $\begin{array}{}{2}_{\mu }^{\ast }-1=r+s,\phantom{\rule{thickmathspace}{0ex}}0 Then utilizing the definition of O1, u1L(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 $\begin{array}{}\underset{\mathit{\Omega }}{\int }\frac{dx}{|x-\left(1-ϵ\right)\sigma {|}^{\frac{s\left(2N\right)\left(N-2\right)}{2N-\mu }}}<\mathrm{\infty }.\end{array}$ 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

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

• Subcase 2

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 $\begin{array}{}{A}_{ϵ}^{2}\end{array}$ is sum of eight non-negative integrals and each integral has an upper bound of the form $\begin{array}{}C\int {\int }_{{O}_{2}}\frac{{u}_{1}\left(x\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{{2}_{\mu }^{\ast }-1}\left({g}_{\rho }^{ϵ,\sigma }\left(y\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}C\int {\int }_{{O}_{2}}\frac{\left({u}_{1}\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}\left(t{g}_{\rho }^{ϵ,\sigma }\left(y\right)\right)\left({u}_{1}\left(x\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy.\end{array}$ By the similar estimates as in Subcase 1, definition of O2, the fact that $\begin{array}{}t{g}_{\rho }^{ϵ,\sigma }\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ 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 $\begin{array}{}\left({u}_{1}\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}=\left({u}_{1}\left(y\right){\right)}^{r}.\left({u}_{1}\left(y\right){\right)}^{s}\end{array}$ with $\begin{array}{}{2}_{\mu }^{\ast }-1=r+s,\phantom{\rule{thickmathspace}{0ex}}0<1+s<\frac{{2}_{\mu }^{\ast }}{2}.\end{array}$ Then utilizing the definition of O2, u1L(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 $\begin{array}{}\underset{\mathit{\Omega }}{\int }\frac{dx}{|x-\left(1-ϵ\right)\sigma {|}^{\frac{\left(1+s\right)\left(2N\right)\left(N-2\right)}{2N-\mu }}}<\mathrm{\infty }.\end{array}$ Hence we obtain

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

• Subcase 3

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 $\begin{array}{}{A}_{ϵ}^{3}\end{array}$ is sum of eight non-negative integrals and each integral has an upper bound of the form $\begin{array}{}C{\iint }_{{O}_{3}}\frac{\left({u}_{1}\left(x\right){\right)}^{{2}_{\mu }^{\ast }-1}\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right)\right)\left({u}_{1}\left(y\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}C{\iint }_{{O}_{3}}\frac{{u}_{1}\left(y\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}\left({g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy.\end{array}$ By the similar estimates as in Subcase 1, Subcase 2, definition of O3 and regularity of u1, we get $\begin{array}{}{A}_{ϵ}^{3}\le O\left({ϵ}^{\left(\frac{2N-\mu }{4}\right)\mathit{\Theta }}\right)\end{array}$ for all Θ < 1.

• Subcase 4

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 $\begin{array}{}{A}_{ϵ}^{4}\end{array}$ is sum of eight non-negative integrals and each integral has an upper bound of the form $\begin{array}{}C\int {\int }_{{O}_{4}}\frac{\left({u}_{1}\left(x\right){\right)}^{{2}_{\mu }^{\ast }-1}\left(t{g}_{\rho }^{ϵ,\sigma }\left(x\right)\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(y\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}C\int {\int }_{{O}_{4}}\frac{{u}_{1}\left(y\right)\left(t{g}_{\rho }^{ϵ,\sigma }\left(y\right){\right)}^{{2}_{\mu }^{\ast }-1}\left({g}_{\rho }^{ϵ,\sigma }\left(x\right){\right)}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy.\end{array}$ 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 $\begin{array}{}{A}_{ϵ}^{i}\le O\left({ϵ}^{\left(\frac{2N-\mu }{4}\right)\mathit{\Theta }}\right)\end{array}$ 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)

where u1 is the local minimum in Lemma 3.10.

#### Proof

By Lemma 3.8, uL(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 $\begin{array}{}\mathit{\Theta }=\frac{2}{{2}_{\mu }^{\ast }},\end{array}$ 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 $\begin{array}{}K\left(t\right):=\frac{{t}^{2}}{2}\left({S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+O\left({ϵ}^{N-2}\right)\right)-\frac{{t}^{{2.2}_{\mu }^{\ast }}}{{2.2}_{\mu }^{\ast }}\left({S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}-O\left({ϵ}^{\frac{2N-\mu }{2}}\right)\right)-{t}^{{2.2}_{\mu }^{\ast }-1}\stackrel{^}{C}{R}_{0}{ϵ}^{\frac{N-2}{2}}\end{array}$ then K(t) → ∞ as t → ∞ and $\begin{array}{}\underset{t\to {0}^{+}}{lim}\end{array}$ K(t) > 0 so there exists a tϵ > 0 such that $\begin{array}{}\underset{t>0}{sup}\end{array}$ K(t) is attained and $\begin{array}{}{t}_{ϵ}<{\left(\frac{{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+O\left({ϵ}^{N-2}\right)}{{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}-O\left({ϵ}^{\frac{2N-\mu }{2}}\right)}\right)}^{\frac{1}{{2.2}_{\mu }^{\ast }-2}}\end{array}$ := 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)

#### Lemma 4.6

The following holds:

1. $\begin{array}{}{H}_{0}^{1}\left(\mathit{\Omega }\right)\setminus {\mathcal{N}}_{f}^{-}={U}_{1}\cup {U}_{2},\end{array}$ where

$U1:=u∈H01(Ω)∖{0}|u+≢0,∥u∥t−u∥u∥.$

2. $\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$U1.

3. For each 0 < ϵϵ0, there exists t0 > 1 and such that u1 + $\begin{array}{}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\end{array}$U2.

4. For each 0 < ϵ < ϵ0, there exists s0 ⊂ (0, 1) and such that $\begin{array}{}{u}_{1}+{s}_{0}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\in {\mathcal{N}}_{f}^{-}.\end{array}$

5. $\begin{array}{}{\mathit{Υ}}_{f}^{-}<{\mathit{Υ}}_{f}+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$

#### Proof

1. It holds by Lemma 3.3 (d).

2. Let u$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$ then t+(u) = 1 and $\begin{array}{}1<{t}^{+}\left(u\right)<{t}_{max}<{t}^{-}\left(u\right)=\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)\end{array}$ that is, $\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$U1.

3. First, we will show that there exists a constant c > 0 such that $\begin{array}{}0<{t}^{-}\left(\frac{{u}_{1}+t{g}_{\rho }^{ϵ,\sigma }}{\parallel {u}_{1}+t{g}_{\rho }^{ϵ,\sigma }\parallel }\right) for all t > 0. On the contrary, let there exist a sequence {tn} such that tn → ∞ and $\begin{array}{}{t}^{-}\left(\frac{{u}_{1}+{t}_{n}{g}_{\rho }^{ϵ,\sigma }}{\parallel {u}_{1}+{t}_{n}{g}_{\rho }^{ϵ,\sigma }\parallel }\right)\to \mathrm{\infty }\end{array}$ as n → ∞. Define $\begin{array}{}{u}_{n}:=\frac{{u}_{1}+{t}_{n}{g}_{\rho }^{ϵ,\sigma }}{\parallel {u}_{1}+{t}_{n}{g}_{\rho }^{ϵ,\sigma }\parallel }\end{array}$ so there exists t(un) such that t(un)un$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$. 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 $\begin{array}{}0<{t}^{-}\left(\frac{{u}_{1}+t{g}_{\rho }^{ϵ,\sigma }}{\parallel {u}_{1}+t{g}_{\rho }^{ϵ,\sigma }\parallel }\right) for all t > 0. Let $\begin{array}{}{t}_{0}=\frac{|{c}^{2}-\parallel {u}_{1}{\parallel }^{2}{|}^{\frac{1}{2}}}{\parallel {g}_{\rho }^{ϵ,\sigma }\parallel }+1\end{array}$ 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 + $\begin{array}{}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\end{array}$U2.

4. For each 0 < ϵ < ϵ0, define a path ξϵ(s) = u1 + $\begin{array}{}s{t}_{0}{g}_{\rho }^{ϵ,\sigma }\end{array}$ for s ∈ [0, 1]. Then

$ξϵ(0)=u1andξϵ(1)=u1+t0gρϵ,σ∈U2.$

Since $\begin{array}{}\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)\end{array}$ is a continuous function and ξϵ([0, 1]) is connected. So, there exists s0 ∈ [0, 1] such that $\begin{array}{}{\xi }_{ϵ}\left({s}_{0}\right)={u}_{1}+{s}_{0}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\in {\mathcal{N}}_{f}^{-}.\end{array}$

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} ⊂ $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) be such that 𝓙f(un) → c, $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(un) → 0. Then passing if necessary to a subsequence, there exists a solution v0$\begin{array}{}{H}_{0}^{1}\end{array}$(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 viD1,2(ℝN) and k sequences $\begin{array}{}\left\{{y}_{n}^{i}{\right\}}_{n}\end{array}$Ω and $\begin{array}{}\left\{{\lambda }_{n}^{i}{\right\}}_{n}\end{array}$ ⊂ 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 $\begin{array}{}{\mathcal{J}}_{\mathrm{\infty }}\left(u\right):=\frac{1}{2}\underset{{\mathbb{R}}^{N}}{\int }|\mathrm{\nabla }u{|}^{2}\text{\hspace{0.17em}}dx-\frac{1}{{2.2}_{\mu }^{\ast }}\underset{{\mathbb{R}}^{N}}{\int }\underset{{\mathbb{R}}^{N}}{\int }\frac{|{u}^{+}\left(x\right){|}^{{2}_{\mu }^{\ast }}|{u}^{+}\left(y\right){|}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy,\phantom{\rule{1em}{0ex}}u\in {D}^{1,2}\left({\mathbb{R}}^{N}\right).\end{array}$

#### Lemma 4.8

1. Let {un} be a (PS)c sequence for 𝓙f with $\begin{array}{}c<{\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ then there exists a subsequence still denoted by {un} and a nonzero u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu0 strongly in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and 𝓙f(u0) = c.

2. Let {un} ⊂ $\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ be a (PS)c sequence for 𝓙f with

$Υf(Ω)+N−μ+22(2N−μ)SH,L2N−μN−μ+2

then there exists subsequence still denoted by {un} and a nonzero u0$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ such that unu0 strongly in $\begin{array}{}{H}_{0}^{1}\end{array}$(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 $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and there exists a subsequence of {un} still denoted by {un} and u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu0 in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(u0) = 0. Then by Lemma 3.11, we have either u0$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ 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, $\begin{array}{}{\mathcal{J}}_{\mathrm{\infty }}\left({v}_{1}\right)=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ 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$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$, 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 YX is contractible to a point in X if and only if there exists a continuous mapping ϱ : [0, 1] × YX such that for some x0X, ϱ(0, x) = x for all xY and ϱ(1, x) = x0 for all xY.

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 GC1(X, ℝ). Assume that there are c1 ∈ ℝ and k ∈ ℕ, such that

1. G satisfies the Palais-Smale condition for energy level cc1;

2. cat({xX | G(x) ≤ c1}) ≥ k.

Then G has at least k critical points in {xX | 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–1X 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$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) there exists a unique t > 0 and a unique t* > 0 such that t u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ and t* u ∈ 𝓝.

#### Lemma 5.4

1. For each uΣ := {u$\begin{array}{}{H}_{0}^{1}\end{array}$(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$\begin{array}{}{H}_{0}^{1}\end{array}$(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 uΣ 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 < ∥fH–1 < e11 then $\begin{array}{}{\mathit{Υ}}_{f}^{-}>0.\end{array}$

#### Proof

1. For each uΣ, define $\begin{array}{}k\left(t\right)=\frac{1}{2}{t}^{2}-\frac{{t}^{{2.2}_{\mu }^{\ast }}}{{2.2}_{\mu }^{\ast }}{b}_{c}\left(u\right),\end{array}$ 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$\begin{array}{}{H}_{0}^{1}\end{array}$(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 uΣ 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 $\begin{array}{}{\mathit{Υ}}_{0}=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}>0\end{array}$ 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 $\begin{array}{}{\mathit{Υ}}_{f}^{-}\end{array}$(Ω) > 0 whenever ∥fH–1 < e11

#### Lemma 5.5

If Ω satisfies condition (A) then there exists a δ0 > 0 such that if u ∈ 𝓝 with $\begin{array}{}\mathcal{J}\left(u\right)\le \frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+\end{array}$ δ0, then $\begin{array}{}\underset{{\mathbb{R}}^{N}}{\int }\frac{x}{|x|}|\mathrm{\nabla }u{|}^{2}\text{\hspace{0.17em}}dx\ne 0\end{array}$

#### Proof

Let {un} ∈ 𝓝 such that $\begin{array}{}\mathcal{J}\left({u}_{n}\right)=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+o\left(1\right)\end{array}$ and $\begin{array}{}\underset{{\mathbb{R}}^{N}}{\int }\frac{x}{|x|}|\mathrm{\nabla }{u}_{n}{|}^{2}\text{\hspace{0.17em}}dx=0.\end{array}$ Since {un} ∈ 𝓝 therefore by Lemma 2.9, {un} is a Palais-Smale sequence of 𝓙 at level $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ 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, $\begin{array}{}{\lambda }_{n}^{1}\end{array}$ ∈ ℝ+, $\begin{array}{}{y}_{n}^{1}\end{array}$Ω. Moreover, if n → ∞ then $\begin{array}{}{\lambda }_{n}^{1}\end{array}$ → 0, $\begin{array}{}\frac{{y}_{n}^{1}}{|{y}_{n}^{1}|}\to {y}_{0}\end{array}$ 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 < ϵϵ0 (defined in Proposition 4.5), define Hϵ : 𝕊N–1$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) as

$Hϵ(σ)=u1+s0t0gρϵ,σ,$

where the function u1 + s0$\begin{array}{}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\end{array}$ 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, $\begin{array}{}{H}_{ϵ}\left(\sigma \right)={u}_{1}+{s}_{0}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\in {\mathcal{N}}_{f}^{-}.\end{array}$ So we only have to prove that 𝓙f(u1 + s0$\begin{array}{}{t}_{0}{g}_{\rho }^{ϵ,\sigma }\end{array}$) ≤ Υf(Ω) + $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}-{\delta }_{ϵ}\end{array}$ for some δϵ > 0. Since by Proposition 4.5,

$supt≥0Jf(u1+tgρϵ,σ)

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 thatfH–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 $\begin{array}{}u\in \left[{\mathcal{J}}_{f}\le {\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\right]\end{array}$ then $\begin{array}{}{\mathcal{J}}_{f}\left(u\right)\le {\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ and u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$, that is, $\begin{array}{}\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)=1.\end{array}$ Since Υf(Ω) < 0 we have $\begin{array}{}{\mathcal{J}}_{f}\left(u\right)\le \frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ So for $\begin{array}{}\frac{u}{\parallel u\parallel }\in \mathit{\Sigma }\end{array}$ there exits a t* > 0 such that $\begin{array}{}\frac{{t}^{\ast }u}{\parallel u\parallel }\in \mathcal{N}\end{array}$ 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 $\begin{array}{}\left(\left(1-\omega {\right)}^{-\frac{2N-\mu }{N-\mu +2}}-1\right)\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}<\frac{{\delta }_{0}}{2}\end{array}$ where δ0 is defined in Lemma 5.5. Now for 0 < ω < ω0 choose e22 such that if ∥fH–1 < e22 then $\begin{array}{}\frac{1}{2\omega \left(1-\omega {\right)}^{\frac{2N-\mu }{N-\mu +2}}}\parallel f{\parallel }_{{H}^{-1}}^{2}<\frac{{\delta }_{0}}{2}.\end{array}$ 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 $\begin{array}{}G:\left[{\mathcal{J}}_{f}\le {\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\right]\to {\mathbb{S}}^{N-1}\end{array}$ 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 andfH–1 < e22, the map

$GoHϵ:SN−1→SN−1$

is homotopic to the identity.

#### Proof

Define $\begin{array}{}\mathcal{K}:=\left\{u\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\setminus \left\{0\right\}|\underset{{\mathbb{R}}^{N}}{\int }\frac{x}{|x|}|\mathrm{\nabla }u{|}^{2}\text{\hspace{0.17em}}dx\ne 0\right\}\end{array}$ 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 $\begin{array}{}\underset{{\mathbb{R}}^{N}}{\int }\frac{x}{|x|}|\mathrm{\nabla }{g}_{\rho }^{ϵ,\sigma }{|}^{2}\text{\hspace{0.17em}}dx\ne 0\end{array}$ for sufficiently small ϵ. Thus, $\begin{array}{}\overline{G}\left({g}_{\rho }^{ϵ,\sigma }\right)\end{array}$ is well defined. Now let y : [s1, s2] → 𝕊N–1 be a regular geodesic between $\begin{array}{}\overline{G}\left({g}_{\rho }^{ϵ,\sigma }\right)\end{array}$ and G(Hϵ(σ)) such that y(s1) = $\begin{array}{}\overline{G}\left({g}_{\rho }^{ϵ,\sigma }\right)\end{array}$ 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)

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 $\begin{array}{}\underset{\lambda \to {1}^{-}}{lim}\end{array}$ ςϵ(λ, σ) = σ and $\begin{array}{}\underset{\lambda \to {\frac{1}{2}}^{-}}{lim}\end{array}$ςϵ(λ, σ) = $\begin{array}{}\overline{G}\left({g}_{\rho }^{ϵ,\sigma }\right)\end{array}$.

• (i)

$\begin{array}{}\underset{\lambda \to {1}^{-}}{lim}\end{array}$ ςϵ(λ, σ) = σ: Indeed

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

then $\begin{array}{}\underset{\lambda \to {1}^{-}}{lim}\end{array}$ ςϵ(λ, σ) = σ.

• (b)

$\begin{array}{}\underset{\lambda \to {\frac{1}{2}}^{-}}{lim}\end{array}$ ςϵ(λ, σ) = $\begin{array}{}\overline{G}\left({g}_{\rho }^{ϵ,\sigma }\right)\end{array}$: Indeed

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

Hence, ςϵC([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 ∥fH–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. LetfH–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 $\begin{array}{}\mathcal{V}:=\left\{u\in {H}_{0}^{1}\left(\mathit{\Omega }\right):\underset{\mathit{\Omega }}{\int }\underset{\mathit{\Omega }}{\int }\frac{|{u}^{+}\left(x\right){|}^{{2}_{\mu }^{\ast }}|{u}^{+}\left(y\right){|}^{{2}_{\mu }^{\ast }}}{|x-y{|}^{\mu }}\text{\hspace{0.17em}}dxdy=1\right\},\phantom{\rule{thickmathspace}{0ex}}{h}_{\rho }^{ϵ,\sigma }\left(x\right)=\frac{{g}_{\rho }^{ϵ,\sigma }\left(x\right)}{\parallel {g}_{\rho }^{ϵ,\sigma }{\parallel }_{NL}}\end{array}$ where $\begin{array}{}{g}_{\rho }^{ϵ,\sigma }\end{array}$ is defined in (4.1).

#### Lemma 6.1

$\begin{array}{}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }_{{D}^{1,2}\left({\mathbb{R}}^{N}\right)}^{2}\to {S}_{H,L}\end{array}$ 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, $\begin{array}{}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }^{2}<{2}^{\frac{N-\mu +2}{2N-\mu }}{S}_{H,L}.\end{array}$

#### Proof

Since we know that $\begin{array}{}\parallel \mathrm{\nabla }{u}_{ϵ}^{\sigma }{\parallel }_{{L}^{2}\left({\mathbb{R}}^{N}\right)}^{2}=\parallel {u}_{ϵ}^{\sigma }{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ and this on using Lemma 4.2 we get $\begin{array}{}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }^{2}\to {S}_{H,L}\end{array}$ as ρ → 0. So there exists a ρ0 such that 0 < ρ < ρ0, we obtain $\begin{array}{}\underset{\sigma \in {\mathbb{S}}^{N-1},ϵ\in \left(0,1\right]}{sup}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }^{2}<{2}^{\frac{N-\mu +2}{2N-\mu }}{S}_{H,L}.\end{array}$

Now for any u$\begin{array}{}{H}_{0}^{1}\end{array}$(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 $\begin{array}{}\underset{ϵ\to 0}{lim}\beta \left({h}_{\rho }^{ϵ,\sigma }\right)=\sigma .\end{array}$

#### Proof

If there exists η > 0 and a sequence ϵn → 0+ such that $\begin{array}{}|\beta \left({h}_{\rho }^{{ϵ}_{n}}\right)-\sigma |\end{array}$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$\begin{array}{}|\beta \left({h}_{\rho }^{{ϵ}_{n}}\right)-\sigma |\end{array}$n → 0+ as ϵn → 0+, a contradiction.□

#### Lemma 6.4

Let m0 = $\begin{array}{}\underset{u\in \mathcal{Q}}{inf}\end{array}$u2 then SH,L < m0.

#### Proof

Obviously SH,Lm0, so let if possible, SH,L = $\begin{array}{}\underset{u\in \mathcal{Q}}{inf}\end{array}$u2 then there exists a sequence {vn} ∈ $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that ∥vnNL = 1, β(vn) = 0, ∥vn2SH,L as n → ∞. Setting $\begin{array}{}{w}_{n}={S}_{H,L}^{\frac{N-2}{2\left(N-\mu +2\right)}}{v}_{n}\end{array}$ we get $\begin{array}{}\parallel {w}_{n}{\parallel }_{NL}^{{2.2}_{\mu }^{\ast }}={S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ and $\begin{array}{}\parallel {w}_{n}{\parallel }^{2}\to {S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ Therefore, $\begin{array}{}\mathcal{J}\left({w}_{n}\right)\to \frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\end{array}$ and 𝓙(wn)(wn) = o(1). Using Lemma 2.9, we obtain {wn} is a Palais-Smale sequence of 𝓙 at level $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ Subsequently, by [19, Theorem 4.4] and Remark 2.8, there exist sequences ynΩ, λn ∈ ℝ+ such that yny0Ω 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

#### Proof

Apparently SH,L$\begin{array}{}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }_{{D}^{1,2}\left({\mathbb{R}}^{N}\right)}^{2}\end{array}$ and we know that SH,L is not attained on a bounded domain. Thus, SH,L < $\begin{array}{}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }_{{D}^{1,2}\left({\mathbb{R}}^{N}\right)}^{2}\end{array}$. Since SH,L < m0, there exists δ0 such that $\begin{array}{}\frac{{S}_{H,L}}{2}+{\delta }_{0}<\frac{{m}_{0}}{2}\end{array}$ and from Lemma 6.1 we know that $\begin{array}{}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }_{{D}^{1,2}\left({\mathbb{R}}^{N}\right)}^{2}\end{array}$SH,L as ϵ → 0. Therefore for δ0 > 0 there exists a ϵ0 > 0 such that $\begin{array}{}\parallel {h}_{\rho }^{ϵ,\sigma }{\parallel }_{{D}^{1,2}\left({\mathbb{R}}^{N}\right)}^{2}\end{array}$ < 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 ϕC1(Y, ℝ). Let A be a compact metric space, A0A be a closed set and yC(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 u0Y 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 $\begin{array}{}F=\left\{q\in C\left({\overline{B}}_{{r}_{0}},\mathcal{V}\right);\phantom{\rule{thickmathspace}{0ex}}{q}_{|\mathrm{\partial }{\overline{B}}_{{r}_{0}}}={h}_{\rho }^{ϵ,\sigma }\right\}\end{array}$ 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 qF, we have q(Br0) ∩ 𝓠 ≠ ∅.

#### Proof

It is enough to show there exist > 0 and σ̃ ∈ 𝕊N–1 such that β(q((1 – )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)σBr0 then q((1 – ϵ)σ) = $\begin{array}{}{h}_{\rho }^{ϵ,\sigma }\end{array}$ 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)σBr0 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 > 0 and σ̃ ∈ 𝕊N–1 such that ψ((1 – )σ̃) = 0 that is, β(q((1 – )σ̃)) = 0.□

Using above Lemma we have $\begin{array}{}{m}_{0}\le \underset{\left(1-ϵ\right)\sigma \in {\overline{B}}_{{r}_{0}}}{sup}\parallel q\left(\left(1-ϵ\right)\sigma \right){\parallel }^{2}\end{array}$ for all qF. 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 $\begin{array}{}\overline{c}<{2}^{\frac{N-\mu +2}{2N-\mu }}{S}_{H,L}\end{array}$ for 0 < ρ < ρ0. Combining all these and using Lemma 6.4 we have

$SH,L(6.1)

In addition, from Lemma 6.5, we get

$c^=sup∂B¯r0∥hρϵ,σ∥2

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) = $\begin{array}{}\underset{t>0}{max}\end{array}$ 𝓙f(tu) = 𝓙f(t(u)u), where t(u) is defined in Lemma 3.3.

#### Lemma 6.8

The following holds:

1. 𝓙̆fC1(𝓥, ℝ) and $\begin{array}{}〈{\stackrel{˘}{\mathcal{J}}}_{f}^{\mathrm{\prime }}\left(u\right),h〉={t}^{-}\left(u\right)〈{\mathcal{J}}_{f}^{\mathrm{\prime }}\left({t}^{-}\left(u\right)u\right),h〉\end{array}$ for all hTu(𝓥) where $\begin{array}{}{T}_{u}\left(\mathcal{V}\right):=\left\{h\in {H}_{0}^{1}\left(\mathit{\Omega }\right)|\underset{\mathit{\Omega }}{\int }\underset{\mathit{\Omega }}{\int }\frac{|{u}^{+}\left(x\right){|}^{{2}_{\mu }^{\ast }}|{u}^{+}\left(y\right){|}^{{2}_{\mu }^{\ast }-1}h\left(y\right)}{|x-y{|}^{\mu }}=0\right\}.\end{array}$

2. If u ∈ 𝓥 is a critical point of 𝓙̆f then t(u)u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ is a critical point of 𝓙f.

3. If {un}n∈ℕ is a (PS)c sequence of 𝓙̆f then {t(un)un}n∈ℕ$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ is a (PS)c sequence for 𝓙f.

#### Proof

1. For every u$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), t(u)u$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ that is, 〈 $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(t(u)u), u〉 = 0 and $\begin{array}{}\frac{{d}^{2}}{d{t}^{2}}{|}_{t={t}^{-}\left(u\right)}{\mathcal{J}}_{f}\left(tu\right)<0.\end{array}$ 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 hTu(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 〈 $\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(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 $\begin{array}{}{t}^{-}\left({u}_{n}\right)>{\left(\frac{\parallel u{\parallel }^{2}}{{2.2}_{\mu }^{\ast }-1}\right)}^{\frac{1}{{2.2}_{\mu }^{\ast }-2}}>{\left(\frac{{S}_{H,L}}{{2.2}_{\mu }^{\ast }-1}\right)}^{\frac{1}{{2.2}_{\mu }^{\ast }-2}}>C\end{array}$ for some C > 0. Since $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) = RunTun(𝓥) so 〈$\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(un), v〉 = 〈$\begin{array}{}{\mathcal{J}}_{f}^{\mathrm{\prime }}\end{array}$(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$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ is a (PS)c sequence for 𝓙f.□

#### Lemma 6.9

If 0 < ρ < ρ0, then $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}<{y}_{0}<\frac{N-\mu +2}{2N-\mu }{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$

#### Proof

For u ∈ 𝓥, solving $\begin{array}{}{\mathcal{J}}^{\mathrm{\prime }}\left(tu\right)=t\phantom{\rule{thickmathspace}{0ex}}a\left(u\right)-{t}^{{2.2}_{\mu }^{\ast }-1}=0\end{array}$ we get t = 0 and $\begin{array}{}t=\left(a\left(u\right){\right)}^{\frac{1}{{2.2}_{\mu }^{\ast }-2}}.\end{array}$ 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

$\begin{array}{}{\stackrel{˘}{\mathcal{J}}}_{f}\left({h}_{\rho }^{ϵ,\sigma }\right)=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+o\left(1\right)\end{array}$ as ϵ → 0.

#### Proof

By Lemma 4.1, $\begin{array}{}{h}_{\rho }^{ϵ,\sigma }\end{array}$ ⇀ 0 in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) as ϵ → 0. On solving

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

we conclude $\begin{array}{}{t}_{f}=\parallel {g}_{\rho }^{ϵ,\sigma }{\parallel }_{NL}+o\left(1\right).\end{array}$ 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 $\begin{array}{}{e}_{0}^{\ast }\end{array}$ > 0 such that if 0 < ∥fH−1 < $\begin{array}{}{e}_{0}^{\ast }\end{array}$,

$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 ∥fH−1 < e1(δ) then

$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 ∥fH−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

Hence for fix small 0 < ϵ < $\begin{array}{}min\left\{\frac{\frac{N-\mu +2}{2N-\mu }{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}-{y}_{0}}{2},{y}_{0}-\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\right\}\end{array}$ such that if ∥fH−1 < $\begin{array}{}{e}_{0}^{\ast }\end{array}$ = 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

That is, $\begin{array}{}{\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}<{y}_{f}<{\mathit{Υ}}_{f}^{-}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$

#### Proposition 6.12

If 0 < ρ < ρ0, 0 < ∥fH−1 < $\begin{array}{}{e}_{0}^{\ast }\end{array}$ (defined in Lemma 6.11) then there exists a critical point u4$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ of 𝓙f with 𝓙f(u4) = yf.

#### Proof

Let $\begin{array}{}c\in \left({\mathit{Υ}}_{f}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}},\phantom{\rule{thickmathspace}{0ex}}{\mathit{Υ}}_{f}^{-}\left(\mathit{\Omega }\right)+\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}\right)\end{array}$ and {un}n∈ℕ is a (PS)c sequence of 𝓙̆f. Then by Lemma 6.8, {t(un)un}n∈ℕ$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ 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, $\begin{array}{}{y}_{f}>{\stackrel{˘}{\mathcal{J}}}_{f}\left({h}_{\rho }^{ϵ,\sigma }\right)=\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}+o\left(1\right)\end{array}$ 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 $\begin{array}{}{\stackrel{˘}{\mathcal{J}}}_{f}^{\mathrm{\prime }}\left({v}_{4}\right)\end{array}$ = 0. Thus by Lemma 6.8, u4 := t(v4) v4$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ 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$\begin{array}{}{\mathcal{N}}_{f}^{+}\end{array}$$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that 𝓙f(u1) = Υf whenever 0 < ∥fH−1 < e00. By Proposition 5.9 we have two more critical point u2, u3$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ of 𝓙f such that in 𝓙f(u2), 𝓙f(u3) < Υf(Ω) + $\begin{array}{}\frac{N-\mu +2}{2\left(2N-\mu \right)}{S}_{H,L}^{\frac{2N-\mu }{N-\mu +2}}.\end{array}$ Therefore we get three positive solutions of (Pf) whenever 0 < ∥fH−1 < e* where e* is defined in Proposition 5.9. Let e** = min{e*, $\begin{array}{}{e}_{0}^{\ast }\end{array}$} then by Proposition 6.12, we get u4$\begin{array}{}{\mathcal{N}}_{f}^{-}\end{array}$ 𝓙f(u4) = yf.

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Accepted: 2019-03-08

Published Online: 2019-08-06

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 803–835, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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