The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent

TL;DR

This paper investigates how the topology of a domain influences the number of positive solutions for a nonlinear elliptic equation involving Hardy-Littlewood-Sobolev critical exponent, using Lusternik-Schnirelman theory.

Contribution

It establishes a link between the domain's topology and the count of positive solutions for a class of Choquard equations with critical exponent.

Findings

Number of positive solutions equals the category of the domain when λ<λ₁.

Results depend on the dimension N and the exponent q.

Proves existence of multiple solutions based on topological complexity.

Abstract

In this article we are concern for the following Choquard equation \[ -\Delta u = \lambda |u|^{q-2}u +\left(\int_\Omega \frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu} dy \right)|u|^{2^*_\mu-2} u \; \text{in}\; \Omega,\quad u = 0 \; \text{ on } \partial \Omega , \] where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=\frac{2N-\mu}{N-2}$ and $q \in [2,2^*)$ where $2^*=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove if $\lambda< \lambda_1$ then problem has $\text{cat}_{\Omega}(\Omega)$ positive solutions whenever $q \in [2,2^*)$ and $N>3 $ or $4<q<6 $ and $N=3$.

Equations

$$-\Delta u=\lambda |u{|}^{q-2}u+\left({\int }_{\Omega }\frac{|u(y){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{2_{\mu }^{\ast }-2}u\text{in}\Omega ,u=0\text{ on }\partial \Omega ,$$

$$(P_{\lambda})\;\left\{\begin{array}[]{rllll}-\Delta u&=\lambda|u|^{q-2}u+\left(\displaystyle\int_{\Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^{*}_{\mu}-2}u\;\text{in}\;\Omega,\\ u&=0\;\text{ on }\partial\Omega,\end{array}\right.$$

$$-\Delta u=\lambda |u{|}^{q-2}u+|u{|}^{2^{\ast }-2}u\text{ in }\Omega ,u=0\text{ on }\partial \Omega .$$

$$-\Delta u+u=\left(\frac{1}{|x{|}^{\mu }}\ast |u{|}^p\right)|u{|}^{p-2}u\text{ in }{\mathbb{R}}^N.$$

$${\int }_{\Omega }{\int }_{\Omega }\frac{|u(x){|}^q|u(y){|}^q}{|x-y{|}^{\mu }}dydx$$

$$-\Delta u=\left({\int }_{\Omega }\frac{|u(y){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{2_{\mu }^{\ast }-2}u\text{in}\Omega ,u=0\text{ on }\partial \Omega ,$$

$$\{x\in {\mathbb{R}}^N,R_1<|x|<R_2\}\subset \Omega ,\{x\in {\mathbb{R}}^N,|x|<R_1\}⊈\overline{\Omega }.$$

$$-\Delta u-\lambda u=\left({\int }_{\Omega }\frac{|u(y){|}^{p_{\epsilon }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{p_{\epsilon }-2}u\text{in}\Omega ,u=0\text{ on }\partial \Omega ,$$

$$\displaystyle Cat_{X}(Y)=\min\bigg{\{}k$$

$$\displaystyle Y_{j}\text{ is contractible to a point in }X\text{ for all }j\text{ and }\displaystyle\displaystyle\cup_{j=1}^{k}Y_{j}=X\bigg{\}}$$

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{f(x)h(y)}{|x-y{|}^{\mu }}dydx\le C(t,r,\mu ,N)\Vert f{\Vert }_{L^t}\Vert h{\Vert }_{L^r}.$$

$$C(t,r,\mu ,N)=C(N,\mu )={\pi }^{\frac{\mu }{2}}\frac{\Gamma (\frac{N}{2}-\frac{\mu }{2})}{\Gamma (N-\frac{\mu }{2})}{\left\{\frac{\Gamma (\frac{N}{2})}{\Gamma (\frac{\mu }{2})}\right\}}^{-1+\frac{\mu }{N}}.$$

$$h(x)=A({\gamma }^2+|x-a{|}^2{)}^{(2N-\mu )/2},$$

$$\displaystyle D^{1,2}(\mathbb{R}^{N})=\bigg{\{}u\in L^{2^{*}}(\mathbb{R}^{N})\;:\;\nabla u\in L^{2}(\mathbb{R}^{N},\mathbb{R}^{N})\bigg{\}},$$

$$\Vert u\Vert ={\left({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}.$$

$$S=\underset{u\in D^{1,2}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\left\{{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx:{\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }}dx=1\right\}.$$

$$S_{H,L}=\underset{u\in D^{1,2}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\left\{{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx:{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^{2_{\mu }^{\ast }}|u(y){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dxdy=1\right\}.$$

$$u=C{\left(\frac{b}{b^2+|x-a{|}^2}\right)}^{\frac{N-2}{2}}$$

$$S=S_{H,L}{\left(C(N,\mu )\right)}^{\frac{N-2}{2N-\mu }}.$$

$$\Vert .{\Vert }_{NL}:={\left({\int }_{\Omega }{\int }_{\Omega }\frac{|.{|}^{2_{\mu }^{\ast }}|.{|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dydx\right)}^{\frac{1}{2\cdot 2_{\mu }^{\ast }}}$$

$$J_{\lambda }(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx-\frac{\lambda }{q}{\int }_{\Omega }|u{|}^{q}dx-\frac{1}{2\cdot 2_{\mu }^{\ast }}{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x){|}^{2_{\mu }^{\ast }}|u(y){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dydx.$$

$${\left({\int }_{\Omega }{\int }_{\Omega }\frac{|u(x){|}^{2_{\mu }^{\ast }}|u(y){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dydx\right)}^{\frac{1}{2_{\mu }^{\ast }}}\le C(N,\mu {)}^{\frac{2N-\mu }{N-2}}\Vert u{\Vert }_{L^{2^{\ast }}}^2.$$

$$\displaystyle\lambda_{1}=\inf_{u\in H_{0}^{1}(\Omega)\setminus\{0\}}\bigg{\{}\int_{\Omega}|\nabla u|^{2}~{}dx\bigg{|}\int_{\Omega}|u|^{2}~{}dx=1\bigg{\}}.$$

$$(\mathcal{Q})\text{ Assume }0<\lambda <{\lambda }_1.\text{ Moreover, }q\in [2,2^{\ast })\text{ and }N>3\text{ OR }4<q<6\text{ and }N=3.$$

$$\displaystyle J_{\lambda}(u)\geq\left\{\begin{array}[]{ll}\frac{1}{2}\left(1-\frac{\lambda}{\lambda_{1}}\right)\|u\|^{2}-\frac{S_{H,L}^{-1}}{2\cdot 2^{*}_{\mu}}\|u\|^{2\cdot 2^{*}_{\mu}},&\text{ if }q=2,\\ \frac{1}{2}\|u\|^{2}-\frac{\lambda S^{\frac{-q}{2}}|\Omega|^{\frac{2^{*}-q}{2^{*}}}}{q}\|u\|^{q}-\frac{S_{H,L}^{-1}}{2\cdot 2^{*}_{\mu}}\|u\|^{2\cdot 2^{*}_{\mu}},&\text{ if }q\in(2,2^{*}).\\ \end{array}\right.$$

$$J_{\lambda }(tu)$$

$$N_{\lambda }^{\Omega }:=\{u\in H_0^1(\Omega )\setminus \{0\}|⟨J_{\lambda }^{\prime }(u),u⟩=0\}.$$

$${\varphi }_u(t)=$$

$${\varphi }_u^{\prime }(t)=t\Vert u{\Vert }^2-\lambda t^{q-1}{\int }_{\Omega }|u{|}^{q}dx-t^{2\cdot 2_{\mu }^{\ast }-1}\Vert u{\Vert }_{NL}^{2\cdot 2_{\mu }^{\ast }}=t{m}_u(t)$$

$$J_{\lambda }(u)=\left(\frac{1}{2}-\frac{1}{q}\right){\int }_{\Omega }|u{|}^{q}dx+\left(\frac{1}{2}-\frac{1}{2\cdot 2_{\mu }^{\ast }}\right)\Vert u{\Vert }_{NL}^{2\cdot 2_{\mu }^{\ast }}>0.$$

$${\theta }_{\lambda }:=\underset{u\in N_{\lambda }^{\Omega }}{\inf }{J}_{\lambda }(u)\text{and}{\hat{\theta }}_{\lambda }:=\underset{u\in H_0^1(\Omega )\setminus \{0\}}{\inf }\underset{t\ge 0}{\sup }{J}_{\lambda }(tu),$$

Full text

[TABLE]

where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^{N}(N\geq 3)$, $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ and $q\in[2,2^{*})$ where $2^{*}=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove if $\lambda<\lambda_{1}$ then problem has $\text{cat}_{\Omega}(\Omega)$ positive solutions whenever $q\in[2,2^{*})$ and $N>3$ or $4<q<6$ and $N=3$.

Key words: Choquard equation, critical exponent, Lusternik-Schnirelman theory.

2010 Mathematics Subject Classification: 49J35, 35A15, 35J60.

1 Introduction

The purpose of this article is to study the existence and multiplicity of solution of the following Choquard equation

[TABLE]

where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^{N}(N\geq 3)$, $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ and $q\in[2,2^{*})$ where $2^{*}=\frac{2N}{N-2}$.

It is not unfamiliar that nonlinear analysis fascinates many researchers. In particular, the study of elliptic equations is more attractive both for theoretical pde’s and real-world applications. There is an ample amount of literature regarding the existence and multiplicity of solutions of the following equation:

[TABLE]

In the pioneering work of Brezis and Nirenberg [7], authors studied the problem (1.1) with $q=2$ for the existence of a nontrivial solution. Then many researchers studied the elliptic equations involving Sobolev critical exponent in bounded and unbounded domains. In [4], Bahri and Coron studied the problem (1.1) in case of $\lambda=0$ and proved the existence of a positive solution when $\Omega$ is not a contractible domain using homology theory. Subsequently, Rey [26] studied critical elliptic problem (1.1) for $q=2$ and proved that there exist at least $\text{cat}_{\Omega}(\Omega)$ solutions in $H_{0}^{1}(\Omega)$ whenever $\lambda$ is sufficiently small. We cite [5, 6, 11, 2, 30] for existence and multiplicity of solutions of elliptic problems using variational methods, with no attempt to provide the complete list. In the framework of fractional Laplacian, the effect of topology on the number of solutions of problems was discussed in [13, 14] and references therein.

Currently, nonlocal equations appealed a substantial number of researchers, especially the Choquard equations. The work on Choquard equations was started with the quantum theory of a polaron model given by S. Pekar [25] in 1954. After that in 1976, in the modeling of a one component plasma, P. Choquard [22] used the following equation with $\mu=1,\;p=2$ and $N=3$:

[TABLE]

For $\mu=1,\;p=2$ and $N=3$, Lieb [22] proved existence, uniqueness of the ground state solution of (1.2) by using symmetric decreasing rearrangement inequalities. With the help of variational methods, Moroz and Schaftingen [23] established the existence of least energy solutions of (1.2) and prove properties about the symmetry, regularity, and asymptotic behavior at infinity of the least energy solutions. For interested readers, we refer [3, 9, 10, 24] and references therein for the work on Choquard equations.

The Hardy-Littlewood-Sobolev inequality (2.1) plays a significant role in the variational formulation of Choquard equations. Observe that the integral

[TABLE]

is well defined if $\frac{2N-\mu}{N}\leq q\leq\frac{2N-\mu}{N-2}=2^{*}_{\mu}$. Choquard equations involving Hardy-Littlewood-Sobolev critical exponent(that is, $q=2^{*}_{\mu}$) provoke the interest of the mathematical community due to the lack of compactness in the embedding $H_{0}^{1}(\Omega)\ni u\mapsto\frac{|u|^{2^{*}_{\mu}}|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\in L^{1}(\Omega\times\Omega)$. In [15], authors used variational methods to prove the existence and multiplicity of positive solutions for the critical Choquard problem involving convex and convex-concave type nonlinearities.

In this spirit, recently in [20] Goel, Rădulescu and Sreenadh, studied the Coron problem for Choquard equation and proved the existence of a positive high energy solution of the following problem

[TABLE]

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}(N\geq 3)$, $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$, $0<\mu<N$ and satisfies the following conditions: There exists constants $0<R_{1}<R_{2}<\infty$ such that

[TABLE]

In [18] Ghimenti and Pagliardini studied the following slightly subcritical Choquard problem

[TABLE]

where $\varepsilon>0,\;\Omega$ is a regular bounded domain of $\mathbb{R}^{N},\;\lambda\geq 0$ and $p_{\varepsilon}=2^{*}_{\mu}-\varepsilon$. Here authors proved that There exists $\overline{\varepsilon}>0$ such that for every $\varepsilon\in(0,\overline{\varepsilon}]$, Problem (1.3) has at least $cat_{\Omega}(\Omega)$ low energy solutions. Moreover, if $\Omega$ is not contractible, there exists another solution with higher energy.

Motivated by all these, in this paper, we study the existence of multiple solutions of the problem $(P_{\lambda})$. Since the geometry of the domain plays an essential role, here we proved that the topology of the domain yields a lower bound on the number of positive solutions. More precisely, we show that the problem $(P_{\lambda})$ has at least $\text{cat}_{\Omega}(\Omega)$ solutions. Here $\text{cat}_{\Omega}(\Omega)$ is the Lusternik-Schirelman category defined as follows

Definition 1.1

Let $X$ be a topological space and $Y$ be a closed set in X Then

[TABLE]

In order to achieve our aim, we used the fact that Lusternik-Schirelman category is invariant under Nehari manifold. Then using the blowup analysis involving the minimizers and the mountain pass Lemma, we show the infimum of the functional associated with $(P_{\lambda})$ over the the Nehari Manifold is achieved. Moreover we define the barycenter mapping associated to Choquard nonlinear term and apply the machinery of barycenter mapping to prove our desired conclusion. With this introduction we will state our main result:

Theorem 1.2

Let $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^{N}(N\geq 3)$ and $q\in[2,2^{*})$ then there exists $0<\Lambda^{*}<\lambda_{1}$ such that for all $\lambda\in(0,\Lambda^{*})$ there exists at least $cat_{\Omega}(\Omega)$ positive solutions of $(P_{\lambda})$ under the following conditions

$q\in[2,2^{*})$* and $N>3$ or* 2. 2.

$4<q<6$* and $N=3$.*

Turning to layout of the article: In Section 2, we give the variational framework and preliminary results. In Section 3, we give the Palais-Smale analysis and existence of a solution of $(P_{\lambda})$. In Section 4, we prove some technical Lemmas and proof Theorem 1.2. Finally in the appendix, we study the behavior of optimizing sequence of the best constant $S_{H,L}$ defined in (2.2).

2 Variational framework and Preliminary results

To study the problem $(P_{\lambda})$ by variational approach we will start with the stating the celebrated Hardy-Littlewood-Sobolev inequality.

Proposition 2.1

[21]**(Hardy-Littlewood-Sobolev Inequality) Let $t,r>1$ and $0<\mu<N$ with $1/t+\mu/N+1/r=2$, $f\in L^{t}(\mathbb{R}^{N})$ and $h\in L^{r}(\mathbb{R}^{N})$. There exists a sharp constant $C(t,r,\mu,N)$ independent of $f,h$, such that

[TABLE]

If $t=r=2N/(2N-\mu)$, then

[TABLE]

Equality holds in (2.1) if and only if $f\equiv(constant)h$ and

[TABLE]

for some $A\in\mathbb{C},0\neq\gamma\in\mathbb{R}$ and $a\in\mathbb{R}^{N}$. $\square$

The Sobolev space $D^{1,2}(\mathbb{R}^{N})$ is defined as

[TABLE]

endowed with the norm

[TABLE]

The best constant for the embedding $D^{1,2}(\mathbb{R}^{N})$ into $L^{2^{*}}(\mathbb{R}^{N})$ (where $2^{*}=\frac{2N}{N-2}$ )is defined as

[TABLE]

Consequently, we define

[TABLE]

Lemma 2.2

[16]** The constant $S_{H,L}$ defined in (2.2) is achieved if and only if

[TABLE]

where $C>0$ is a fixed constant , $a\in\mathbb{R}^{N}$ and $b\in(0,\infty)$ are parameters. Moreover,

[TABLE]

Lemma 2.3

[16]** For $N\geq 3$ and $0<\mu<N$. Then

[TABLE]

defines a norm on $L^{2^{*}}(\Omega)$, where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^{N}$.

The energy functional associated with $(P_{\lambda})$, $J_{\lambda}:H_{0}^{1}(\Omega)\rightarrow\mathbb{R}$ is defined by

[TABLE]

Employing the Hardy-Littlewood-Sobolev inequality (2.1), we have

[TABLE]

It implies the functional $J_{\lambda}\in C^{1}(H_{0}^{1}(\Omega),\mathbb{R})$. We know that there exists a one to one correspondence between the critical points of $J_{\lambda}$ and solution of $(P_{\lambda})$.

Notation We denote $\lambda_{1}$ be the first eigenvalue of $-\Delta$ with zero Dirichlet boundary data, which is given by

[TABLE]

We also denote $\mathcal{(Q)}$ as the following condition:

[TABLE]

Lemma 2.4

Assume $N\geq 3$ and $\lambda\in(0,\lambda_{1})$. Then $J_{\lambda}$ satisfies the following conditions:

1. (i)

There exists $\alpha,\rho>0$ such that $J_{\lambda}(u)\geq\alpha$ for $\|u\|=\rho$ 2. (ii)

There exists $e\in H_{0}^{1}(\Omega)$ with $\|e\|>\rho$ such that $J_{\lambda}(e)<0$.

**Proof. ** (i) Using Hölder’s inequality, Sobolev inequality and Hardy-Littlewood-Sobolev inequality, we have

[TABLE]

Using the given assumption on $\lambda$ and the fact that $2<2\cdot 2^{*}_{\mu}$, we can choose $\alpha,\rho>0$ such that $J_{\lambda}(u)\geq\alpha$ whenever $\|u\|=\rho$.

(ii) Let $u\in H_{0}^{1}(\Omega)$ then

[TABLE]

Hence we can choose $t_{0}>0$ such that $e:=t_{0}u$ such that (ii) follows. $\square$

The Nehari manifold associated to $J_{\lambda}$ defined as

[TABLE]

Lemma 2.5

Let $u$ be a critical point on $N_{\lambda}^{\Omega}$. Then $u$ is a critical point of $J_{\lambda}$ on $H_{0}^{1}(\Omega)$.

**Proof. ** The proof follows from [12]. $\square$

Lemma 2.6

Assume $\lambda\in(0,\lambda_{1})$. Then $N_{\lambda}^{\Omega}\not=\emptyset$ and $J_{\lambda}$ is bounded below on $N_{\lambda}^{\Omega}$.

**Proof. ** Let $u\in H_{0}^{1}(\Omega)\setminus\{0\}$. Consider the function

[TABLE]

Then $\phi_{u}(t)=0,\;\phi_{u}(t)\rightarrow-\infty$ as $t\rightarrow\infty$. We now show that there exists unique $t_{0}>0$ such that $\phi_{u}^{\prime}(t_{0})=0$ . Since

[TABLE]

where $m_{u}(t)=\|u\|^{2}-b_{u}(t)$ and $b_{u}(t)=\lambda t^{q-2}\displaystyle\int_{\Omega}|u|^{q}~{}dx+t^{2\cdot 2^{*}_{\mu}-2}\|u\|_{NL}^{2\cdot 2^{*}_{\mu}}$. Observe that $b_{u}$ is a continuous function, $\displaystyle\lim_{t\rightarrow\infty}b_{u}(t)=\infty$ and $b_{u}^{\prime}(t)>0$ for all $t>0$. Therefore, there exists unique $t_{0}>0$ such that $b_{u}(t_{0})=\|u\|^{2}$. That is, $\phi_{u}^{\prime}(t_{0})=0$. It implies $t_{0}\phi_{u}^{\prime}(t_{0})=0$ and $t_{0}u\in N_{\lambda}^{\Omega}$. It implies $N_{\lambda}^{\Omega}\not=\emptyset$. Now if $u\in N_{\lambda}^{\Omega}$ then $J_{\lambda}(u)$ reduced to

[TABLE]

Therefore, $\displaystyle\inf_{u\in N_{\lambda}^{\Omega}}J_{\lambda}(u)>0$. That is, $J_{\lambda}$ is bounded below on $N_{\lambda}^{\Omega}$. $\square$

Now we set

[TABLE]

where $\widehat{\theta}_{\lambda}$ denote the Mountain Pass (MP, in short) level.

3 The Palais-Smale condition and estimates of the functional

In this section we will give the Palais–Smale analysis and prove the existence of a minimizer of the functional $J_{\lambda}$ over the Nehari manifold.

Lemma 3.1

Let $N\geq 3,\;\lambda\in(0,\lambda_{1})$ and $q\in[2,2^{*})$. Then the functional $J_{\lambda}$ satisfies the $(PS)_{c}$ condition for all $c<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$.

**Proof. ** Let $\{u_{n}\}$ be a sequence in $H_{0}^{1}(\Omega)$ such that

[TABLE]

**Claim 1: ** $u_{n}$ is a bounded sequence in $H_{0}^{1}(\Omega)$.

On the contrary assume that $\|u_{n}\|\rightarrow\infty$. Let $\widetilde{u_{n}}=\displaystyle\frac{u_{n}}{\|u_{n}\|}$ be a sequence in $H_{0}^{1}(\Omega)$ then $\|\widetilde{u_{n}}\|=1$ for all $n$. Therefore we can assume there exists $\widetilde{u}$, up to subsequences

[TABLE]

Using (3.1) we have

[TABLE]

It implies that

[TABLE]

Now if $q>2$ and $\lambda>0$ then by the assumption $\|u_{n}\|\rightarrow\infty$, we get $\|\widetilde{u_{n}}\|\rightarrow\infty$, which is not possible. If $q=2$ and $\lambda\in(0,\lambda_{1})$, then $0<\left(1-\frac{\lambda}{\lambda_{1}}\right)\|u_{n}\|^{2}\leq o_{n}(1)$, which is again not possible, this concludes the proof of Claim.

Hence we can assume, there exists a $u_{0}\in H_{0}^{1}(\Omega)$ such that up to a subsequence $u_{n}\rightharpoonup u_{0}$ weakly in $H_{0}^{1}(\Omega)$, $u_{n}\rightarrow u_{0}\text{ strongly in }L^{r}(\Omega)\text{ for all }r\in[1,2^{*})$ and $u_{n}\rightarrow u_{0}$ a.e. on $\Omega$. Using all this and proceeding with the same assertions as in [16, Lemma 2.4], we get $J_{\lambda}^{\prime}(u_{0})=0$. Now the Brezis-Leib Lemma (See [8, 16]) leads to

[TABLE]

and

[TABLE]

It implies $J_{\lambda}(u_{0})+\frac{N-\mu+2}{2(2N-\mu)}\|u_{n}-u_{0}\|^{2}=c+o_{n}(1)$ and if $\|u_{n}-u_{0}\|^{2}\rightarrow M$ as $n\rightarrow\infty$ then by (3.2), $\|u_{n}-u_{0}\|_{NL}^{2\cdot 2^{*}_{\mu}}\rightarrow M$ as $n\rightarrow\infty$. If $M=0$ then we are done otherwise if $M>0$ then using the definition of $S_{H,L}$, we have $M^{\frac{1}{2^{*}_{\mu}}}S_{H,L}\leq M$ that is, $S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}\leq M$. Since $\langle J_{\lambda}^{\prime}(u_{0}),\;u_{0}\rangle=0$, it gives

[TABLE]

Resuming the information collected so far, what we have gained is that,

[TABLE]

which yields a contradiction to the range of $c$. Hence compactness of the sequence follows. $\square$

Lemma 3.2

Let $N\geq 3$ and $\lambda\in(0,\lambda_{1})$ then $J_{\lambda}$ constraint to $N_{\lambda}^{\Omega}$ satisfies the $(PS)_{c}$ condition for all $c<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$.

**Proof. ** Let $u_{n}\in N_{\lambda}^{\Omega}$ be such that $J_{\lambda}(u_{n})\rightarrow c$ and there exists a sequence $\{\alpha_{n}\}$ in $\mathbb{R}$ with

[TABLE]

where the functional $T_{\lambda}$ is defined as $T_{\lambda}(u)=\|u\|^{2}-\lambda\int_{\Omega}|u|^{q}~{}dx-\|u\|_{NL}^{2\cdot 2^{*}_{\mu}}$. First of all, we will show that $u_{n}$ is a bounded sequence in $H_{0}^{1}(\Omega)$. From the fact that $J_{\lambda}(u_{n})\rightarrow c$, it is easy to see that there exists a positive constant $C_{1}$ such that $|J_{\lambda}(u_{n})|<C_{1}$. If $q\in(2,2^{*})$ then using the fact that $u_{n}\in N_{\lambda}^{\Omega}$, we deduce that

[TABLE]

If $q=2$, for $\lambda\in(0,\lambda_{1})$, we obtain, for any $n\in\mathbb{N}$,

[TABLE]

This proves that $u_{n}$ is a bounded sequence in $H_{0}^{1}(\Omega)$. It implies that $\{\langle T_{\lambda}^{\prime}(u_{n}),u_{n}\rangle\}$ is a bounded sequence in $\mathbb{R}$ and there exists $\kappa\in(-\infty,0]$ such that, up to a subsequence, $\langle T_{\lambda}^{\prime}(u_{n}),u_{n}\rangle\rightarrow\kappa$ as $n\rightarrow\infty$. Let if possible, $\kappa<0$ then using the fact that $u_{n}\in N_{\lambda}^{\Omega}$ and (1.5), we have

[TABLE]

This implies $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. That is,

[TABLE]

which on employing Lemma 3.1 gives that $u_{n}$ has a convergent subsequence. At last suppose $\kappa=0$. Since

[TABLE]

then $\int_{\Omega}|u_{n}|^{q}~{}dx\rightarrow 0$ and $\|u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}\rightarrow 0$. Taking into account the fact $u_{n}\in N_{\lambda}^{\Omega}$ we have $\|u_{n}\|\rightarrow 0$. That is, $u_{n}\rightarrow 0$ strongly in $H_{0}^{1}(\Omega)$. $\square$

In order to proceed further we will use the minimizer of $S_{H,L}$. From Lemma 2.2 we know that

[TABLE]

are the minimizers of $S_{H,L}$. Without loss of generality, let us assume that $0\in\Omega$. This implies there exists a $\delta>0$ such that $B_{4\delta}(0)\subset\Omega$. Now define $\eta\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $0\leq\eta\leq 1$ in $\mathbb{R}^{N}$, $\eta\equiv 1$ in $B_{\delta}(0)$ and $\eta\equiv 0$ in $\mathbb{R}^{N}\setminus B_{2\delta}(0)$ and $|\nabla\eta|<C$. Let $u_{\varepsilon}\in H_{0}^{1}(\Omega)$ be defined as $u_{\varepsilon}(x)=\eta(x)U_{\varepsilon}(x)$.

Proposition 3.3

Let $N\geq 3,\;0<\mu<N$ and $q\in(2,2^{*})$ then the following holds:

1. (a)

$\|u_{\varepsilon}\|^{2}\leq S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}+O(\varepsilon^{N-2})$. 2. (b)

$\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}\leq S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}+O(\varepsilon^{N})$* and $\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}\geq S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}-O(\varepsilon^{N})$.* 3. (c)

\displaystyle\int_{\Omega}|u_{\varepsilon}|^{2}~{}dx\geq C\left\{\begin{array}[]{ll}\varepsilon^{2}+O(\varepsilon^{N-2}),&\text{ if }N>4,\\ \varepsilon^{2}|\log\;\varepsilon|+O(\varepsilon^{2}),&\text{ if }N=4\\ \varepsilon^{N-2}+O(\varepsilon^{2}),&\text{ if }N<4.\\ \end{array}\right.** 4. (d)

$\displaystyle\int_{\Omega}|u_{\varepsilon}|^{q}~{}dx\geq O(\varepsilon^{N-\frac{N-2}{2}q})$* whenever $q\in(2,2^{*})$ and $N>3$ OR $4<q<6$ and $N=3$.*

**Proof. ** For (a) and (c) See [29, Lemma 1.46]. For (b) See [19, Proposition 2.8]. For (d), first let $N>3$ and $2<q<2^{*}$ then $0<(N-2)q-N<N$. Now let $N=3$ and $4<q<6$ then $1<q-3<3$. Hence we have the following estimate

[TABLE]

$\square$

Lemma 3.4

Let $N\geq 3$ and $\lambda>0$ and condition $\mathcal{(Q)}$ holds. Then $\widehat{\theta}_{\lambda}<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$.

**Proof. ** By the definition of $\widehat{\theta}_{\lambda}$, it is enough to show that for $u_{\varepsilon}\in H_{0}^{1}(\Omega)$,

[TABLE]

Let

[TABLE]

then using the same assertions as in Lemma 2.6 for the function $\mathcal{G}$, we deduce that there exists unique $t_{\varepsilon}>0$ such that $\displaystyle\sup_{t\geq 0}\mathcal{G}(t)=\mathcal{G}(t_{\varepsilon})=J_{\lambda}(t_{\varepsilon}u_{\varepsilon})$ and $\mathcal{G}^{\prime}(t_{\varepsilon})=0$, provided $\lambda\in(0,\lambda_{1})$. As a result, we obtain

[TABLE]

It implies $\|u_{\varepsilon}\|^{2}=\lambda t_{\varepsilon}^{q-2}\int_{\Omega}|u_{\varepsilon}|^{q}~{}dx+t_{\varepsilon}^{2\cdot 2^{*}_{\mu}-2}\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}$. Therefore, using Proposition 3.3, Sobolev embedding, definition of $S_{H,L}$ and the fact that $\lambda\in(0,\lambda_{1})$, we deduce

[TABLE]

for some suitable constants $C_{1},C_{2}>0$. It gives that there exists a $T_{1}>0$ such that $t_{\varepsilon}\geq T_{1}$. Also, from (3.4), $t_{\varepsilon}^{2\cdot 2^{*}_{\mu}}\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}\leq t_{\varepsilon}^{2}\|u_{\varepsilon}\|^{2}$. That is,

[TABLE]

Hence

[TABLE]

where $\mathcal{V}(t)=\displaystyle\frac{t^{2}}{2}\|u_{\varepsilon}\|^{2}-\frac{t^{2\cdot 2^{*}_{\mu}}}{2\cdot 2^{*}_{\mu}}\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}$. Now using proposition 3.3 and the fact that $\mathcal{V}(t)$ has maximum at $t^{*}=\left(\frac{\|u_{\varepsilon}\|^{2}}{\|u_{\varepsilon}\|_{NL}^{2\cdot 2^{*}_{\mu}}}\right)^{\frac{1}{2\cdot 2^{*}_{\mu}-2}}$, we get

[TABLE]

**Case 1: ** $N>3$ and $q\in(2,2^{*})$ OR $N=3$ and $4<q<6$.

As a consequence of Proposition 3.3 and (3.5), we have

[TABLE]

Now using the condition of $N$ and $q$, we have $N-\frac{N-2}{2}q<N-2$ then for $\varepsilon$ sufficiently small, $C_{1}\varepsilon^{N-2}-\frac{\lambda T_{1}^{q}}{q}C_{2}\varepsilon^{N-\frac{N-2}{2}q}<0$. Therefore,

[TABLE]

**Case 2: ** If $q=2$ and $N>3$.

When $N>4$ then by Proposition 3.3 and (3.5),

[TABLE]

Therefore, for $\varepsilon$ sufficiently small, $C_{1}\varepsilon^{N-2}-\frac{\lambda T_{1}^{2}}{2}C_{2}\varepsilon^{2}<0$, we obtain

[TABLE]

When $N=4$ then again by Proposition 3.3 and (3.5), for an appropriate constant $C_{3}>0$, we have

[TABLE]

Since $|\log\varepsilon|\rightarrow\infty$ as $\varepsilon\rightarrow 0$, for $\varepsilon$ sufficiently small, $C_{3}\varepsilon^{2}-\frac{\lambda T_{1}^{2}}{2}C_{2}\varepsilon^{2}|\log\varepsilon|<0$. Thus

[TABLE]

$\square$

Lemma 3.5

If condition $\mathcal{(Q)}$ holds then the following holds.

1. (a)

$\widehat{\theta}_{\lambda}=\theta_{\lambda}$. 2. (b)

$0<\theta_{\lambda}<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$. 3. (c)

There exists $u_{\lambda}^{\Omega}\in N_{\lambda}^{\Omega}$ such that $J_{\lambda}(u_{\lambda}^{\Omega})=\displaystyle\inf_{u\in N_{\lambda}^{\Omega}}J_{\lambda}(u)=\theta_{\lambda}$ and $u_{\lambda}^{\Omega}\geq 0$.

**Proof. **

1. (a)

By Lemma 3.1, Lemma 3.4, Lemma 2.4 and Mountain Pass Lemma, there exists a $u_{\lambda}^{\Omega}\in H_{0}^{1}(\Omega)$ such that $J_{\lambda}(u_{\lambda}^{\Omega})=\widehat{\theta}_{\lambda}$ and $J_{\lambda}^{\prime}(u_{\lambda}^{\Omega})=0$. It implies $u_{\lambda}^{\Omega}\in N_{\lambda}^{\Omega}$. Hence, $\theta_{\lambda}\leq J_{\lambda}(u_{\lambda}^{\Omega})=\widehat{\theta}_{\lambda}$. Also from Lemma 2.6, for each $v\in N_{\lambda}^{\Omega}$, there exists a unique $t_{0}>0$ such that $\displaystyle\sup_{t\geq 0}J_{\lambda}(tv)=J_{\lambda}(t_{0}v)$. Since $u_{\lambda}^{\Omega}\in N_{\lambda}^{\Omega}$, it implies $\widehat{\theta}_{\lambda}\leq\displaystyle\sup_{t\geq 0}J_{\lambda}(tu)=J_{\lambda}(u)$. Therefore, $\widehat{\theta}_{\lambda}\leq\theta_{\lambda}$. 2. (b)

By Lemma 2.6, $\theta_{\lambda}>0$ and by Lemma 3.4, $\theta_{\lambda}=\widehat{\theta}_{\lambda}<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$. 3. (c)

By part (a), there exists a $u_{\lambda}^{\Omega}\in N_{\lambda}^{\Omega}$ such that $J_{\lambda}(u_{\lambda}^{\Omega})=\widehat{\theta}_{\lambda}=\theta_{\lambda}=\displaystyle\inf_{u\in N_{\lambda}^{\Omega}}J_{\lambda}(u)$. Since $J_{\lambda}(u_{\lambda}^{\Omega})=J_{\lambda}(|u_{\lambda}^{\Omega}|)$, we can assume $u_{\lambda}^{\Omega}\geq 0$. $\square$

4 Proof of Theorem 1.2

In this section, first we gather some information which is needed to estimate the $\text{cat}_{\Omega}(\Omega)$. Before that, we prove some Lemmas which are necessary for the proof of Theorem 1.2.

Lemma 4.1

Let $N\geq 3$ and $\{u_{n}\}$ be a sequence in $H_{0}^{1}(\Omega)$ such that

[TABLE]

Then, there exist sequences $z_{n}\in\mathbb{R}^{N}$ and $\alpha_{n}\in\mathbb{R}^{+}$ such that the sequence

[TABLE]

have a convergent subsequence, still denoted by $v_{n}$. Moreover, $v_{n}\rightarrow v\not\equiv 0$ in $D^{1,2}(\mathbb{R}^{N}),\;z_{n}\rightarrow z\in\overline{\Omega}$ and $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$.

**Proof. ** Let $\{w_{n}\}$ be a sequence such that $w_{n}=\displaystyle\frac{u_{n}}{\|u_{n}\|_{NL}}$ then $\|w_{n}\|_{NL}=1,\;\|w_{n}\|^{2}=\displaystyle\frac{\|u_{n}\|^{2}}{\|u_{n}\|_{NL}^{2}}=\|u_{n}\|^{2(\frac{N-\mu+2}{2N-\mu})}\leq S_{H,L}+o_{n}(1)$. By definition of $S_{H,L}$, $\|w_{n}\|^{2}\geq S_{H,L}$, it implies $\|w_{n}\|^{2}\rightarrow S_{H,L}$ as $n\rightarrow\infty$. Now using Proposition A.1 for the sequence $\{w_{n}\}$, we have the desired result. $\square$

Since $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$, thus we can pick $\delta>0$ small enough so that

[TABLE]

are homotopically equivalent to $\Omega$. Without loss of generality, we can assume that $B_{\delta}=B_{\delta}(0)\subset\Omega$. Consequently, we consider the functional $J_{\lambda}^{B_{\delta}}:H_{0,\text{rad}}^{1}(B_{\delta})\rightarrow\mathbb{R}^{N}$ defined as

[TABLE]

where $H_{0,\text{rad}}^{1}(B_{\delta})=\{u\in H_{0}^{1}(B_{\delta}):u\text{ is radial}\}.$ And let $N_{\lambda}^{B_{\delta}}$ be the Nehari manifold associated to functional $J_{\lambda}^{B_{\delta}}$. Then all the results obtained in Section 3 are valid for the functional $J_{\lambda}^{B_{\delta}}$. In particular, by Lemma 3.5, we know that there exists $u_{\lambda}^{B_{\delta}}\in N_{\lambda}^{B_{\delta}}$ such that $u_{\lambda}^{B_{\delta}}\geq 0$ in $B_{\delta}$. Moreover,

[TABLE]

Now with the help of $u_{\lambda}^{B_{\delta}}$ we will define the following set

[TABLE]

and the function $\phi_{\lambda}:\Omega_{\delta}^{-}\rightarrow\mathcal{A}_{\lambda}$ given by

[TABLE]

In the succession, we define the barycenter mapping $\beta:N_{\lambda}^{\Omega}\rightarrow\mathbb{R}^{N}$ by setting

[TABLE]

Using the fact that $u_{\lambda}^{B_{\delta}}$ is radial, $\beta(\phi_{\lambda}(z))=z$ for all $z\in\Omega_{\delta}^{-}$.

Lemma 4.2

Let $N\geq 3$ and $q\in[2,2^{*})$. Then there exists $\Upsilon^{*}>0$ such that if $u\in\mathcal{A}_{\lambda}$ and $\lambda\in(0,\Upsilon^{*})$ then $\beta(u)\in\Omega_{\delta}^{+}$.

**Proof. ** On the contrary, let there exists sequences $\{\lambda_{n}\}\in\mathbb{R}^{+}$ and $u_{n}\in\mathcal{A}_{\lambda_{n}}$ such that $\lambda_{n}\rightarrow 0$ and $\beta(u_{n})\not\in\Omega_{\delta}^{+}$. Using the definition of $\mathcal{A}_{\lambda_{n}}$, we have $u_{n}\in N_{\lambda_{n}}^{\Omega}$ and $J_{\lambda_{n}}(u_{n})\leq J_{\lambda_{n}}^{B_{\delta}}(u_{\lambda_{n}}^{B_{\delta}})$. Define

[TABLE]

using the same assertions and arguments as in Lemma 2.6, there exists a unique $t_{0}>0$ such that $M^{\prime}(t_{0})=0$ and $t_{0}u_{n}\in N_{\lambda_{n}}^{\Omega}$. Since $u_{n}\in N_{\lambda_{n}}^{\Omega}$, it implies that $M^{\prime}(1)=0$ and $M$ is increasing for $t<1$ and decreasing $t>1$. Therefore,

[TABLE]

As $\|u_{n}\|^{2}-\lambda_{n}\int_{\Omega}|u_{n}|^{q}~{}dx-\|u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}=0$, employing this with definition of $S_{H,L}$ and Sobolev embedding, we have

[TABLE]

where $c_{1}>0$ is a appropriate constant. It implies that for large $n$, there exists a constant $C>0$ such that

[TABLE]

**Claim 1: ** There exists a $l>0$ such that up to a subsequence $\|u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}\rightarrow l$ as $n\rightarrow\infty$.

Since $J_{\lambda_{n}}(u_{n})\leq J_{\lambda_{n}}^{B_{\delta}}(u_{\lambda_{n}}^{B_{\delta}})<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$, $J_{\lambda_{n}}(u_{n})$ is bounded in $\mathbb{R}$, subsequently $\|u_{n}\|_{NL}$ is a bounded sequence. Moreover, from the fact that $u_{n}\in N_{\lambda_{n}}^{\Omega}$, it follows that

[TABLE]

It implies that $\lambda_{n}\int_{\Omega}|u_{n}|^{q}~{}dx$ is a bounded sequence. As a consequence, $\|u_{n}\|$ is bounded in $\mathbb{R}$. Therefore, there exists a $l\geq 0$ such that $\|u_{n}\|_{NL}\rightarrow l$ as $n\rightarrow\infty$. To prove the Claim 1, it is enough to show that $l\not=0$. Using (4.7), we deduce

[TABLE]

where $c_{2}>0$ is a suitable constant. Since $\lambda_{n}\rightarrow 0$, so we have $l>0$. This proves Claim 1.

Claim 2: For all $n\in\mathbb{N}$, there exists $t_{n}>0$ such that $\|t_{n}u_{n}\|^{2}=\|t_{n}u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}$. Furthermore, $t_{n}$ is a bounded sequence in $\mathbb{R}$.

Assume $t_{n}=\left[\frac{\|u_{n}\|^{2}}{\|u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}}\right]^{\frac{1}{2\cdot 2^{*}_{\mu}-2}}$ then $\|t_{n}u_{n}\|^{2}=\|t_{n}u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}\text{ for all }n\in\mathbb{N}$. Using the fact that $\|u_{n}\|$ is bounded and by Claim 1, we deduce that $t_{n}$ is a bounded sequence in $\mathbb{R}$, concludes the proof of Claim 2.

By the definition of $J_{\lambda_{n}}$ and taking into account (4.1), (4.6), Claim 2, $u_{n}\in\mathcal{A}_{\lambda_{n}}$, $\lambda_{n}\rightarrow 0$, and $\int_{\Omega}|u_{n}|^{q}~{}dx$ is bounded, we obtain

[TABLE]

From Claim 2 and Lemma 4.1, there exists a sequences $z_{n}\in\mathbb{R}^{N}$ and $\alpha_{n}\in\mathbb{R}^{+}$ such that the sequence

[TABLE]

have a convergent subsequence, still denoted by $v_{n}$. Moreover, $v_{n}\rightarrow v\not\equiv 0$ in $D^{1,2}(\mathbb{R}^{N}),\;z_{n}\rightarrow z\in\overline{\Omega}$ and $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $\psi(x)=x$ for all $x\in\overline{\Omega}$. Consider

[TABLE]

where the last one follows from regularity of $\psi$ and Lebesgue dominated theorem. This contradicts the assumption $\beta(u_{n})\not\in\Omega_{\delta}^{+}$. It concludes the proof. $\square$

Lemma 4.3

Assume $N\geq 3,\;q\in[2,2^{*})$ and $\lambda\in(0,\Upsilon^{*})$ (defined in Lemma 4.2). Then $\text{cat}_{\mathcal{A}_{\lambda}}(\mathcal{A}_{\lambda})\geq\text{cat}_{\Omega}(\Omega)$

**Proof. ** The proof can be done by using the same assertions as in [2, Lemma 4.3]. $\square$

Next we need following lemma in order to proof Theorem 1.2.

Lemma 4.4

[1]** Suppose that $X$ is a Hilbert manifold and $F\in C^{1}(X,\mathbb{R})$ . Assume that there are $c_{1}\in\mathbb{R}$ and $k\in\mathbb{N}$, such that

$F$* satisfies the Palais-Smale condition for energy level $c\leq c_{1}$;* 2. 2.

$\text{Cat}(\{x\in X\;|\;F(x)\leq c_{1}\})\geq k$.

Then $F$ has at least $k$ critical points in $\{x\in X\;|\;F(x)\leq c_{1}\}$.

Proof of Theorem 1.2 : By Lemma 3.2, $J_{\lambda}$ satisfies $(PS)_{c}$ condition on $N_{\lambda}^{\Omega}$ for any $c<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$, provided $\lambda\in(0,\lambda_{1})$. If condition $\mathcal{(Q)}$ holds then from Lemma 3.5, $0<\theta_{\lambda}<\frac{N-\mu+2}{2(2N-\mu)}S_{H,L}^{\frac{2N-\mu}{N-\mu+2}}$. Hence if condition $\mathcal{(Q)}$ holds then Lemmas 4.3 and 4.4, we have at least $\text{cat}_{\Omega}(\Omega)$ critical points of $J_{\lambda}$ restricted to $N_{\lambda}$ for any $\lambda\in(0,\Lambda^{*})$, where

[TABLE]

Thus using Lemma 2.5, we obtain $J_{\lambda}$ has at least $\text{cat}_{\Omega}(\Omega)$ critical points on $H_{0}^{1}(\Omega)$. From [15, Lemma 4.4], we have at least $\text{cat}_{\Omega}(\Omega)$ positive solutions of problem $(P_{\lambda})$. $\square$

Appendix A ​​​​​​ppendix

Here we will proof behavior of the optimizing sequence of $S_{H,L}$. For the local case, Proposition A.1 has been proved in [28] and [29]. Combining the ideas of [17] and [29], one expects the Proposition A.1 to hold for critical Choquard case, but as best of our knowledge this type of result has not been proved exclusively anywhere. For $N=3$, Proposition A.1 has been proved in [27].

Proposition A.1

Let $\{u_{n}\}$ be a sequence in $H_{0}^{1}(\Omega)$ such that

[TABLE]

Then, there exists a sequences $z_{n}\in\mathbb{R}^{N}$ and $\alpha_{n}\in\mathbb{R}^{+}$ such that the sequence

[TABLE]

have a convergent subsequence, still denoted by $v_{n}$, such that $v_{n}\rightarrow v\not\equiv 0$ in $D^{1,2}(\mathbb{R}^{N}),\;z_{n}\rightarrow z\in\overline{\Omega}$, and $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. In particular, $v$ is a minimizer of $S_{H,L}$.

**Proof. ** Define the Lévy concentration function

[TABLE]

It is easy to see that for each $n,\;\displaystyle\lim_{\lambda\rightarrow 0^{+}}Q_{n}(\lambda)=0$ and $\displaystyle\lim_{\lambda\rightarrow\infty}Q_{n}(\lambda)=1$, there exists $\alpha_{n}>0$ such that $Q_{n}(\alpha_{n})=\frac{1}{2}$. Also, there exist $z_{n}\in\mathbb{R}^{N}$ such that

[TABLE]

Now define the function $v_{n}(x)=\alpha_{n}^{\frac{N-2}{2}}u_{n}(\alpha_{n}x+z_{n})$ then

[TABLE]

It implies $\{v_{n}\}$ is a bounded sequence in $D^{1,2}(\mathbb{R}^{N})$. Therefore, there exist a subsequence, still denoted by $\{v_{n}\}$ such that $v_{n}\rightharpoonup v$ weakly in $D^{1,2}(\mathbb{R}^{N})$, for some $v\in D^{1,2}(\mathbb{R}^{N})$. Then we can assume that there exist $\omega,\;\tau,\;\nu$ such that

[TABLE]

Now using the Brezis-Leib lemma in sense of measure, we have

[TABLE]

Moreover, if we define

[TABLE]

then by using concentration-compactness principle [17, Lemma 2.5], we deduce that

[TABLE]

Also, if $v=0$ and $\displaystyle\int_{\mathbb{R}^{N}}d\omega=S_{H,L}\left(\displaystyle\int_{\mathbb{R}^{N}}d\nu\right)^{\frac{1}{2^{*}_{\mu}}}$ then $\nu$ is concentrated at a single point. By using [17, (2.11)], we have

[TABLE]

It implies

[TABLE]

Using the definition of $S_{H,L}$, (1.2) and (LABEL:lu10), we obtain

[TABLE]

Thanks to the fact that $\|v\|_{NL},\;\displaystyle\int_{\mathbb{R}^{N}}d\kappa,\;\nu_{\infty}$ are non-negative, we get $\|v\|_{NL},\;\displaystyle\int_{\mathbb{R}^{N}}d\kappa,\;\nu_{\infty}$ are equal to either 1 or 0. Using (A), we have $\nu_{\infty}\leq\frac{1}{2}$. It implies $\nu_{\infty}=0$. Now if $\displaystyle\int_{\mathbb{R}^{N}}d\kappa=1$ then $\|v\|_{NL}=0$ that is, $v=0$ a.e. on $\mathbb{R}^{N}$. Therefore, $S_{H,L}=\displaystyle\int_{\mathbb{R}^{N}}d\varpi+\omega_{\infty}\geq\displaystyle\int_{\mathbb{R}^{N}}d\varpi$. Hence,

[TABLE]

Coupling (1.2), (1.4) with the fact that $v=0$ a.e on $\mathbb{R}^{N}$, we have $\nu$ is concentrated at a single point $z_{0}$. From (A), we get

[TABLE]

which is not possible. Hence, $\|v\|^{2\cdot 2^{*}_{\mu}}_{NL}=1$. Also, $S_{H,L}=\displaystyle\lim_{n\rightarrow\infty}\|\nabla v_{n}\|^{2}_{L^{2}}=\|\nabla v\|^{2}_{L^{2}}$. In particular, $v$ is a minimizer of $S_{H,L}$. From [16, Lemma 1.2], we know $S_{H,L}$ is achieved if and only if

[TABLE]

where $C>0$ is a fixed constant, $a\in\mathbb{R}^{N}$ and $b\in(0,\infty)$ are parameters. It implies $v=u=C\left(\frac{b}{b^{2}+|x-a|^{2}}\right)^{\frac{N-2}{2}}$. In particular, $v\not\equiv 0$. Now, we will prove that $\alpha_{n}\rightarrow 0$ and $z_{n}\rightarrow z_{0}\in\overline{\Omega}$. Let if possible $\alpha_{n}\rightarrow\infty$. Since $\{u_{n}\}$ is a bounded sequence in $H_{0}^{1}(\Omega)$, $\{u_{n}\}$ is a bounded sequence in $L^{2}(\Omega)$. Thus if we define $\Omega_{n}=\displaystyle\frac{\Omega-z_{n}}{\alpha_{n}}$ then

[TABLE]

Contrary to this, by Fatou’s Lemma we have $0=\displaystyle\liminf_{n\rightarrow\infty}\int_{\Omega_{n}}|v_{n}|^{2}~{}dx\geq\int_{\Omega_{n}}|v|^{2}~{}dx$. This means $v\equiv 0$, which is not true. Hence $\{\alpha_{n}\}$ is bounded in $\mathbb{R}$ that is, there exists $\alpha_{0}\in\mathbb{R}$ such that $\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$. If $z_{n}\rightarrow\infty$ then for any $x\in\Omega$ and large $n$, $\alpha_{n}x+z_{n}\not\in\overline{\Omega}$. Since $u_{n}\in H_{0}^{1}(\Omega)$ then $u_{n}(\alpha_{n}x+z_{n})=0$ for all $x\in\Omega$, it yields a contradiction to the assumption $\|u_{n}\|_{NL}^{2\cdot 2^{*}_{\mu}}=1$. Therefore, $z_{n}$ is bounded, it implies that $z_{n}\rightarrow z_{0}$. Now suppose $\alpha_{n}\rightarrow\alpha_{0}>0$ then $\Omega_{n}\rightarrow\displaystyle\frac{\Omega-z_{0}}{\alpha_{0}}=\Omega_{0}\not=\mathbb{R}^{N}$. Hence

[TABLE]

which is not true. Hence $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. Finally, arguing by contradiction, we assume that

[TABLE]

In view of the fact that $\alpha_{n}x+z_{n}\rightarrow z_{0}$ for all $x\in\Omega$ as $n\rightarrow\infty$. Now using (1.5) we have $\alpha_{n}x+z_{n}\not\in\overline{\Omega}$ for all $x\in\Omega$ and $n$ large enough. It implies that $u_{n}(\alpha_{n}x+z_{n})=0$ for $n$ large enough. This yields a contradiction, therefore, $z_{0}\in\overline{\Omega}$. $\square$

Acknowledgment The author would like to thank Prof. K Sreenadh for various discussion that greatly improved the manuscript. The author would like to thank the anonymous referee for valuable comments.