Existence of Ground State Solutions for Choquard Equation with the Upper Critical Exponent

Abstract

In this article, we investigate the existence of a nontrivial solution for the nonlinear Choquard equation with upper critical exponent see Equation (6). The Riesz potential in this case has never been studied. We establish the existence of the ground state solution within bounded domains $\mathsf{\Omega }\subset {\mathbb{R}}^N$. Variational methods are used for this purpose. This method proved to be instrumental in our research, enabling us to address the problem effectively. The study of the existence of ground state solutions for the Choquard equation with a critical exponent has applications and relevance in various fields, primarily in theoretical physics and mathematical analysis.

1. Introduction

The Choquard equation

$$-∆u+u=\left(I_{\alpha }\mathrm{\ast }{\left|u\right|}^{\zeta }\right){\left|u\right|}^{\zeta -2}u.$$

(1)

arises in numerous fields of physics (quantum mechanics [1], and modelling of one-component plasma [2], self-gravitating matter [3]). In recent years, several physical contests have proposed the Choquard equations and have attracted a great deal of interest. Choquard first introduced the Choquard equation in the modelling of one-component plasma in 1976. Recently, many scholars have paid attention to following the Choquard equation with local nonlinear perturbation and variable potential:

$$\left\{\begin{array}{l}-∆u+V\left(x\right)u=\left(I_{\alpha }\ast {\left|u\right|}^{\zeta }\right){\left|u\right|}^{\zeta -2}u+g\left(u\right), in {\mathbb{R}}^N;\\ u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right), \end{array}\right.$$

(2)

where $1+\frac{\alpha }{N}<\zeta <\frac{N+\alpha }{N-\alpha },$ $I_{\mathit{\alpha }}:{\mathbb{R}}^N\to \mathbb{R}$ is a Riesz potential of order $\alpha \in (0,N)$in the Euclidean space ${\mathbb{R}}^N$ of dimension $N\ge 3$, which is defined by

$$I_{\alpha }\left(x\right)=\frac{\Gamma \left(\frac{N-\alpha }{2}\right)}{2^{\alpha }\Gamma \left(\frac{\alpha }{2}\right){\pi }^{\frac{N}{2}}{\left|x\right|}^{N-\alpha }},$$

and $\Gamma$ is the Gamma function. Nonlinearity is described by the exponent $\zeta \in \mathbb{R}$.

Equation (2) is used in many different contexts and has different physical meanings. For example, Pekar [4] studied the interaction of free electrons within the lattice of ions with phonons related to lattice distortions by using the conditions if $\mathfrak{g}=0$ and $V\left(x\right)\equiv 1.$In Ref. [2], Lieb studied the positive solution for problem (2) in ${\mathbb{R}}^3$ by using the conditions if $\mathfrak{g}=0,$ $V\left(x\right)\equiv 1, \alpha =2$and $\zeta =2$. In Ref. [5], Tang et al. examined ground state solutions employing the Nehari-type approach for the Choquard equation with a lower critical exponent. They applied specific conditions, including $\mathfrak{g}={\lambda \left|u\right|}^{P-2}u,$ $V\left(x\right)\equiv 1, \alpha \in \left(0,N\right), \zeta =\frac{\alpha }{N}-1$, $\lambda >0, 2<P<2^{*} and N\ge 1.$In Ref. [6], Jing et al. explored the existence of ground state solutions in the context of Choquard equations that involved a localized nonlinear disturbance, employing the Pohožaev identity and minimax methodologies. Lü [7] proved the concentration results and the existence of ground state solutions when $\mu ⟶+\infty$ for problem (2) with subcritical exponents, and using the condition if $\mathfrak{g}=0,$ the potential is well described by $V\left(x\right)=1+\mu h\left(x\right)$ and $N=3$.

Moroz and Van [8] demonstrated that the following general Choquard equation

$$-∆u+u=\left(I_{\alpha }\mathrm{\ast }F\left(u\right)\right)f\left(u\right), in {\mathbb{R}}^N,$$

(3)

has the ground state solution if the following assumptions are met for nonlinearity $f$.

❖

$f\in C(\mathbb{R},\mathbb{R}).$

❖

$\lambda >0$ exists such that for every $s\in \mathbb{R},\left|sf\left(s\right)\right|\le \lambda \left({\left|s\right|}^{\frac{\alpha }{N}+1}+{\left|s\right|}^{\frac{N+\alpha }{N-2}}\right).$

❖

$\underset{s\to 0}{\lim }\frac{F\left(s\right)}{{\left|s\right|}^{(N+\alpha )/N}}=\underset{s\to \mathrm{\infty }}{\lim }\frac{F\left(s\right)}{{\left|s\right|}^{(N+\alpha )/(N-2)}}=0$, where $F\left(s\right)={\int }_0^{s}f\left(t\right)dt.$

❖

$s_0\in \mathbb{R}\backslash \left\{0\right\}$ exists such that $F\left(s_0\right)\ne 0.$

Gui and Tang [9] showed that the following problem has ground state solutions

$$-∆u+u=(I_{\alpha }\mathrm{\ast }{\left|u\right|}^{\frac{N+\alpha }{N-2}}){\left|u\right|}^{\frac{N+\alpha }{N-2}-2}u+\mathfrak{g}(u), in {\mathbb{R}}^N,$$

(4)

if the following conditions are met for $\mathfrak{g}:$

❖

function $\mathfrak{g}\in C(\mathbb{R},\mathbb{R})$ is odd,

❖

$\underset{s\to 0}{\lim }\frac{\mathfrak{g}\left(s\right)}{s}=\underset{s\to \infty }{\lim }\frac{\mathfrak{g}\left(s\right)}{s^{2^{*}-1}}=0,$ where $2^{*}=\frac{2N}{N-2}.$

❖

$\underset{s\to \infty }{\lim }\frac{\mathit{G}\left(s\right)}{s^2}=+\infty$for $N\ge 5;$

❖

$\underset{s\to \infty }{\lim }\frac{\mathit{G}\left(s\right)}{s^2\ln s}=+\infty$for $N=4;$

❖

$\underset{s\to \infty }{\lim }\frac{\mathit{G}\left(s\right)}{s^4}=+\infty$for $N=3$, where $G(x)={\int }_0^s\mathfrak{g}\left(t\right)dt.$

When $\mathfrak{g}=0$, there is no solution to problem (4), as stated in reference [10]. The incorporation of non-relativistic Newtonian gravity into the Schrödinger equation resulted in the transformation of the Choquard equation into what is now referred to as the Schrödinger–Newton equation.

Our work is associated with the following Choquard$–$Pekar equation

$$-∆u+V\left(x\right)u=\left(\frac{1}{{\left|x\right|}^{\mu }}\mathrm{\ast }{\left|u\right|}^{\zeta }\right){\left|u\right|}^{\zeta -2}u in {\mathbb{R}}^N,$$

(5)

In 1954 [1], Pekar studied the case $\zeta =2$ and $\mu =1.$ This situation is related to an explanation of the quantum theory of the stationary state of the polaron. In a particular approximation of the Hartree–Fock theory for one-component plasma, Choquard formulated a model representing an electron ensnared within its self-contained influence in 1977. A Schrödinger$-$Newton equation, first introduced by Penrose in the context of the self-gravitational collapse of the quantum mechanical wave function, is an alternate name for this equation. Lieb established the uniqueness and existence, up to translations, of a ground state to Equation (5) in 1977 [2]. In 1980, Lions [11] investigated the existence of a sequence of solutions that exhibit radial symmetry in Equation (5). The generalized Choquard Equation (5) for $\zeta \ge 2,$ was introduced by Ma and Zhao in 2010 [12], with the assumption that a specific set of real numbers, determined by $\zeta ,$ $N,$and $\mu ,$ is nonempty. Every positive solution of (5) has been proven to be monotonically decreasing and radially symmetric about some points. In 2012, the authors in [13] proved multiplicity results and existence in the electromagnetic case under some assumptions. They also obtained some asymptotical decay and the regularity of the ground states of Equation (5) as they approach infinity. In 2013, Moroz and Van [10] proved the positivity, regularity and radial symmetry of ground states within an optimal range of parameters. Additionally, they derived the asymptotic decay of these states as they approach infinity.

Now, we will study the cases of $V$for Equation (5).

• In the case where $V$ is the continuous periodic function and $\underset{{\mathbb{R}}^N}{\inf }V\left(x\right)>0,$ applying a mountain pass theorem can easily prove the existence result, and a nonlocal term is invariant under translation, see [14].
• The problem is strongly indefinite if $V,$ the periodic potential, changes its sign and the point $0$ lies in the gap of the spectrum of a Schrödinger operator $-∆+V$. The existence of a nontrivial solution with $F\left(u\right)=u^2$ and $\mu =1$ was proven in [15] via the reduction approach.
• When $V$ is a vanishing potential, the studies that highlighted a generalized Choquard equation with the vanishing potential include [16,17].
• When $V>0$ and for some $\gamma \in \left[\mathrm{0,1}\right),$ $\underset{\left|x\right|\to \infty }{\lim }\mathit{inf}V\left(x\right){\left|x\right|}^{\gamma }>0.$ S. Secchi [16] replaced $∆$by ${\epsilon }^2∆,$ and, thus, achieved the existence result for small $\epsilon$ by using a Lyapunov $-$Schmidt-type reduction.
• In the case where $V\left(\infty \right)=0$ and potential $V$ is vanishing at infinity, the author in [18] showed the Choquard Equation (5) with potentials vanishing at infinity. In Ref. [19], Berestycki and Lions published work that refers to this class of problems.

Benci, Grisanti and Micheletti [20,21] considered the following problem

$$\left\{\begin{array}{c}-∆u+V\left(x\right)u=f\left(u\right), in {\mathbb{R}}^N; \\ u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right), \end{array}\right.$$

and proved the nonexistence and existence of a ground state solution by assuming some conditions for nonlinearity $f$. In 2022, the author in Ref. [22] used a generalized Nehari manifold technique developed by Szulkin and Weth to establish the existence of a ground state solution for a class of Choquard equations with a potential. They presumptively considered that a potential satisfies a general indefinite periodic condition. Thus, the Schrödinger operator has a purely continuous spectrum, and the energy function connected to it is strongly indefinite. For a few decades, variational approaches have been used to study qualitative properties and the existence of the Choquard equation. In recent years, there has been thorough research into the presence of a ground state solution for the Choquard equation, as documented in references [23,24,25,26,27,28,29,30,31]. For a more comprehensive understanding of variational methods, readers are encouraged to consult the following references [32,33,34,35].

In this article, we consider the Choquard equation as follows:

$$\left\{\begin{array}{c}-∆u+V\left(x\right)u=\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\mathrm{\ast }{\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{\alpha -N+4}{N-2}}u, in {\mathbb{R}}^N,\\ u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right), \end{array}\right.$$

(6)

where $N\in \mathbb{N},N\ge 3,$ ${∆}^D$ is a Laplacian with Dirichlet boundary conditions on Ω. The exponent $\frac{N+\alpha }{N-2}$ is critical with respect to an upper Hardy–Littlewood–Sobolev inequality, $V\left(x\right)\in L^{\infty }\left({\mathbb{R}}^N\right)$ and ${\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}$ represents a Riesz potential of order $\alpha \in (0,N)$ defined for all $x\in {\mathbb{R}}^{\mathit{N}}\backslash \left\{0\right\}$ by

$${\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}=C\left(N,\alpha \right)\left(P.V{\int }_{\mathsf{\Omega }}\frac{u\left(x\right)-u\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy+{\int }_{{\mathbb{R}}^N\backslash \mathsf{\Omega }}\frac{u\left(x\right)-g\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy\right),$$

where $P.V$ represents the principal value of an integral:

$$P.V{\int }_{{\mathbb{R}}^N}\frac{u\left(x\right)-u\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy=\underset{\epsilon \to 0}{\lim }{\int }_{{\mathbb{R}}^N\backslash B_{\epsilon }\left(x\right)}\frac{u\left(x\right)-u\left(y\right)}{{\left|x-y\right|}^{N-\alpha }},$$

where $B_{\epsilon }\left(x\right)$ is the ball of radius $\epsilon$ and centre $x$.

$$C\left(N,\alpha \right)=\frac{\mathsf{\Gamma }\left(\frac{N-\alpha }{2}\right)}{2^{-\alpha }{\pi }^{(N\backslash 2)}\left|\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)\right|}.$$

The focus of this article lies in the following modification. We replace the Riesz potential $I_{\alpha }$ with (−${∆}^D$)^−α/2, where ${∆}^D$ denotes the Laplacian operator acting on $\mathsf{\Omega }$ with Dirichlet boundary conditions. Previous research had examined the Riesz potential $I_{\alpha }$ as (−$∆$)^−α/2, where $∆$ stands for the Laplacian on $\mathsf{\Omega }$.

We focus on a solution to Choquard Equation (6), which corresponds to critical points of a functional $\mathcal{J}$ defined for each function $u:{\mathbb{R}}^N\to \mathbb{R}$ by

$$\mathcal{J}\left(u\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left|\nabla u\right|}^2+V{\left|u\right|}^2\right)-\frac{N-2}{2(N+\alpha )}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{\alpha +N}{N-2}}\right){\left|u\right|}^{\frac{\alpha -N+4}{N-2}}.$$

(7)

The key concept underpinning our proof begins with demonstrating, firstly, the existence of a nontrivial critical point for the functional $\mathcal{J}$ through the application of the mountain pass lemma and the concentration compactness argument. Following this, we proceed to seek a minimizer for the subsequent minimization problem

$${\delta }_0 :=\inf \left\{\left.\mathcal{J}\left(u\right)\right| u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right)\backslash \left\{0\right\} and \acute{\mathcal{J}}\left(u\right)=0\right\}.$$

The present paper is structured as follows. In the second section, we provide necessary definitions, theorems, lemmas and a crucial estimate related to the mountain pass energy level. Moving on to the third section, we examine the functional $\mathcal{J}$ with a mountain pass geometry, allowing us to obtain the Palais–Smale sequence. Subsequently, we establish an estimate for the mountain pass energy level to ensure compactness. Afterward, we prove the existence of ground state solutions under consideration of the Riesz potential. Following that, we demonstrate the existence of a minimizer. Finally, we address the case of nonexistence.

2. Preliminaries

We present some rudimentary notations in this part, and we will utilize the following:

The usual Sobolev space is ${\mathsf{H}}^1\left({\mathbb{R}}^N\right)$.

We define the inner product in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right)$ by

$$\left(u,v\right)={\int }_{{\mathbb{R}}^N}\left(\nabla u\nabla v+V\left(x\right)uv\right)dx, u,v\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right).$$

We define a norm in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right)$ by

$${‖u‖}^2={\int }_{{\mathbb{R}}^N}\left({\left|\nabla u\right|}^2+V\left(x\right){\left|u\right|}^2\right)dx, u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right).$$

We define the norm in ${\mathrm{D}}^{\mathrm{1,2}}\left({\mathbb{R}}^N\right),$ and (${\mathrm{D}}^{\mathrm{1,2}}\left({\mathbb{R}}^N\right)$is a completion of ${\mathcal{C}}_0^{\infty }\left({\mathbb{R}}^N\right)$)by

$${{‖u‖}^2}_{{\mathrm{D}}^{\mathrm{1,2}}\left({\mathbb{R}}^N\right)}={\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{2}dx,$$

A Palais$-$Smale sequence: Let $\mathcal{F}$ be the real Hilbert space, and $I:\mathcal{F}⟶\mathbb{R}$ is the functional for class $C^1$. If $u_n$ satisfies

$$I\left(u_n\right)⟶c \mathrm{and} I^{\prime }\left(u_n\right)⟶0, \mathrm{as} n⟶\infty .$$

We refer that $u_n\subset \mathcal{F}$ is the Palais$-$Smale sequence at level $c$ for $I$, also known as the ${\left(\mathrm{P}\mathrm{S}\right)}_c$ sequence. If any ${\left(\mathrm{P}\mathrm{S}\right)}_c$ sequence possesses the convergent subsequence, then we assume that $I$ satisfies the ${\left(\mathrm{P}\mathrm{S}\right)}_c$ condition.

The Hardy–Littlewood–Sobolev inequality ([34], Theorem 4.5): We assume that $\mathfrak{B}\in L^s\left({\mathbb{R}}^N\right)$ and $\mathfrak{g}{\in L}^t\left({\mathbb{R}}^N\right)$. Then, we have

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{\mathfrak{B}\left(x\right)\mathfrak{g}\left(y\right)}{{\left|x-y\right|}^{\alpha }}dxdy\le C\left(s,t,\alpha \right){‖\mathfrak{B}‖}_s{‖\mathfrak{g}‖}_t,$$

(8)

where $1<s,t<\mathrm{\infty },0<\alpha <N,$ and $\frac{1}{s}+\frac{1}{t}+\frac{\alpha }{N}=2$.

The sharp constant satisfies

$$C\left(N,\alpha ,s\right)\le \frac{N}{N-\alpha }{\left(\frac{\left|{\mathbb{S}}^{N-1}\right|}{N}\right)}^{\frac{\alpha }{N}}\frac{1}{st}\left({\left(\frac{\frac{\alpha }{N}}{1-\frac{1}{s}}\right)}^{\frac{\alpha }{N}}+{\left(\frac{\frac{\alpha }{N}}{1-\frac{1}{t}}\right)}^{\frac{\alpha }{N}}\right).$$

(9)

If $s=t=\frac{2N}{2N-\alpha },$ then

$$C\left(N,\alpha ,s\right)=C\left(N,\alpha \right)={\pi }^{\frac{\alpha }{2}}\frac{\mathsf{\Gamma }\left(\frac{N}{2}-\frac{\alpha }{2}\right)}{\mathsf{\Gamma }\left(N-\frac{\alpha }{2}\right)}{\left\{\frac{\mathsf{\Gamma }\left(\frac{N}{2}\right)}{\mathsf{\Gamma }\left(N\right)}\right\}}^{-1+\frac{\alpha }{N}}.$$

In this instance, equality in (8) exists if and only if $\mathfrak{g}\equiv \left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}.\right)\mathfrak{B}$ and

$$\mathfrak{B}\left(x\right)=\kappa {({\gamma }^2+{\left|x-a\right|}^2)}^{\frac{-(2N-\alpha )}{2}},$$

for some $\kappa \in \mathbb{C},0\ne \gamma \in \mathbb{R}$ and $a\in {\mathbb{R}}^N.$

Lemma 1 

([34]). We assume that $N\ge 1, \alpha \in (0,N)$ and $s\in \left(1,\frac{N}{\alpha }\right).$ If $\phi \in L^s\left({\mathbb{R}}^N\right),$ then ${\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}\ast \phi \in L^{\frac{Ns}{N-\alpha s}}\left({\mathbb{R}}^N\right),$ and

$${\int }_{{\mathbb{R}}^N}{\left|{\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}*\phi \right|}^{\frac{Ns}{N-\alpha s}}\le C{\left({\int }_{{\mathbb{R}}^N}{\left|\phi \right|}^s\right)}^{\frac{N}{N-\alpha s}},$$

(10)

where $C$ represents a constant that relies exclusively on $\alpha ,N$ and $s.$ $C>0.$

By a semi-group identity for the Riesz potential ${\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}={\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{4}}\ast {\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{4}}$ ([34], Corollary 5.10), we can rewrite the Hardy–Littlewood–Sobolev inequality (10) as

$${\int }_{{\mathbb{R}}^N}\left({\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}\ast {\left|u\right|}^p\right){\left|u\right|}^p={\int }_{{\mathbb{R}}^N}{\left|{\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{4}}*{\left|u\right|}^p\right|}^2\le C{\left({\int }_{{\mathbb{R}}^N}{\left|u\right|}^2\right)}^p.$$

(11)

The Hardy–Littlewood–Sobolev inequality (10) in our situation is

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\le {C\left({\int }_{{\mathbb{R}}^N}{\left|u\right|}^2\right)}^{\frac{N+\alpha }{N-2}},$$

(12)

where a constant $C$ depends on $\alpha ,N.$ The minimization problem for this inequality is as follows:

$$\aleph =inf\left\{\left.{\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right|u\in D^{\mathrm{1,2}}\left({\mathbb{R}}^N\right) and {\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}=1\right\}>0.$$

(13)

By using ([34], Theorem 4.3), the infimum $\mathrm{\aleph }$ is obtained by the function $u\in H^1\left({\mathbb{R}}^{\mathit{N}}\right)$ if and only if the following exists for all $x\in {\mathbb{R}}^{\mathit{N}}$

$$u_{\epsilon }\left(x\right)=A{\left(\frac{{\epsilon }^{\frac{1}{2}}}{\epsilon +{\left|x\right|}^2}\right)}^{\frac{N-2}{2}},$$

(14)

for a few given constants $\epsilon \in \left(0,+\mathrm{\infty }\right),$ $a\in {\mathbb{R}}^N$ and $A\in \mathbb{R}.$

Definition 1 

([35]). The function $u\in H_{loc}^1\left({\mathbb{R}}^N\right)$ is the weak solution to the Choquard Equation (1). For all test functions $\psi \in H^1\left({\mathbb{R}}^N\right)$ supported in the compact set and $\left(I_{\alpha }\ast {\left|u\right|}^{\zeta }\right){\left|u\right|}^{\zeta -2}u$ $\in H_{loc}^{-1}\left({\mathbb{R}}^N\right)$, we have

$${\int }_{{\mathbb{R}}^N}\nabla u·\nabla \psi +u\psi ={\int }_{{\mathbb{R}}^N}\left(I_{\alpha }\ast {\left|u\right|}^{\zeta }\right){\left|u\right|}^{\zeta -2}u\psi .$$

(15)

Definition 2 

((Riesz fractional Laplacian) [36]). To define a fractional Laplacian on the bounded domain $\Omega$ , one approach is to use a real space formula

$${\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}=C\left(N,\alpha \right)\left(P.V{\int }_{\Omega }\frac{u\left(x\right)-u\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy\right),$$

(16)

to functions on $\Omega$. This derivation leads to a Riesz fractional Laplacian in $\Omega$. We discuss Dirichlet boundary conditions here. For defining the Riesz Laplacian within  $\Omega$, Formula (16) requires values of $u$on all of ${\mathbb{R}}^N$. Therefore, the exterior boundary condition

$$u=g in {\mathbb{R}}^N\backslash \mathsf{\Omega },$$

(17)

is necessary. For the functions $u$ that satisfy (17), a Riesz fractional Laplacian is defined for x∈$\Omega$ by

$${\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}u\left(x\right)=C\left(N,\alpha \right)\left(P.V{\int }_{\mathsf{\Omega }}\frac{u\left(x\right)-u\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy+{\int }_{{\mathbb{R}}^N\backslash \mathsf{\Omega }}\frac{u\left(x\right)-g\left(y\right)}{{\left|x-y\right|}^{N-\alpha }}dy\right).$$

The Riesz fractional Laplacian is directly influenced by exterior boundary values  $g$ and the region $\mathsf{\Omega }.$

Lemma 2 

([37], Lemma (10)). Let $\underset{\left|x\right|\to \infty }{\mathit{lim}}infV\left(x\right)\ge 1$ and $V\left(x\right)\in L^{\infty }\left({\mathbb{R}}^N\right).$ If a sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is converged to  $u$ in $L_{loc}^2\left({\mathbb{R}}^N\right)$ and bounded in $L^2\left({\mathbb{R}}^N\right)$, then

$$\underset{n\to \mathit{\infty }}{\lim }inf{\int }_{{\mathbb{R}}^N}V{\left|u_n\right|}^2-{\int }_{{\mathbb{R}}^N}{\left|u_n-u\right|}^2\ge {\int }_{{\mathbb{R}}^N}V{\left|u\right|}^2.$$

Lemma 3 

([10], Lemma (2.4)). We assume that $\alpha \in \left(0,N\right)$ and $N\ge 3.$ If a sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is converged to $u$ nearly everywhere in ${\mathbb{R}}^N$ and the bounded sequence in $L^2\left({\mathbb{R}}^N\right)$, then

$$\begin{array}{c}\underset{n\to \infty }{\mathit{lim}}{\int }_{{\mathbb{R}}^N}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}} \\ \hspace{1em}\hspace{1em}\hspace{1em}=\underset{n\to \infty }{\mathit{lim}}{\int }_{{\mathbb{R}}^N}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}\\ +\underset{n\to \mathit{\infty }}{\mathit{lim}}{\int }_{{\mathbb{R}}^{\mathit{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}\mathit{\ast }{\left|u\right|}^{\frac{\mathit{N}+\mathit{\alpha }}{\mathit{N}-2}}\right){\left|u\right|}^{\frac{\mathit{N}+\alpha }{\mathit{N}-2}}.\end{array}$$

(18)

3. Main Result

We consider that a functional $\mathcal{J}$ has a mountain pass geometry to obtain the Palais–Smale sequence for a functional $\mathcal{J}$.

Theorem 1. 

If a functional $\mathcal{J}$ satisfies the following conditions

(1)

$\sigma >0$ exists such that$\underset{‖u‖=\sigma }{\mathit{inf}}\mathcal{J}\left(u\right)>0;$

(2)

$\underset{t\to +\mathrm{\infty }}{\lim }\mathcal{J}\left(tu\right)=-\mathrm{\infty }$ for any $u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right)\backslash \left\{0\right\},$

then a functional $\mathcal{J}$ has the mountain pass geometry.

Proof. 

(1)

Let ${\mu }_1$ be a constant and ${\mu }_1>0$such that we have the following for $t\in \mathbb{R}$ by using a classical Sobolev inequality and the Hardy–Littlewood$-$Sobolev inequality (11)

$$\begin{array}{c}\mathcal{J}\left(u\right)=\frac{1}{2}{‖u‖}^2-\frac{N-2}{2(N+\alpha )}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\\ \ge \frac{1}{2}{‖u‖}^2-{{\mu }_1‖u‖}^{\frac{2(N+\alpha )}{N-2}}\\ \ge {‖u‖}^2\left(\frac{1}{2}-{\mu }_1{‖u‖}^{\frac{\left(N+\alpha \right)}{N-2}}\right)\\ \underset{‖u‖=\sigma }{\mathit{inf}}\mathcal{J}\left(u\right)\ge {\sigma }^2\left(\frac{1}{2}-{\mu }_1{‖u‖}^{\frac{(N+\alpha )}{N-2}}\right)\\ \underset{‖u‖=\sigma }{\mathit{inf}}\mathcal{J}\left(u\right)\ge {\sigma }^2\left(\frac{1}{2}-{\mu }_1{‖u‖}^{\frac{\left(N+\alpha \right)}{N-2}}\right)>0.\end{array}$$

Then, $\underset{‖u‖=\sigma }{\mathit{inf}}\mathcal{J}\left(u\right)>0$ as long as $\sigma$ is sufficiently small.

(2)

For any non-zero $\mathrm{u}\in H^1\left({\mathbb{R}}^N\right)$ and any positive value of $t,$ we have

$$\mathcal{J}\left(tu\right)\le \frac{t^2}{2}{‖u‖}^2-\frac{\left(N-2\right)t^{\frac{2(N+\alpha )}{N-2}}}{2\left(N+\alpha \right)} {\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}.$$

$\underset{t\to +\infty }{\lim }\mathcal{J}\left(tu\right)=-\infty .$ □

Now, we prove an estimate on a mountain pass energy level to ensure compactness. By using a classical mountain pass theorem, we have the minimax description at an energy level $k_0,$ which is defined by

$$k_0=\underset{\gamma \in \mathsf{\Gamma }}{\mathit{inf}}\underset{t\in \left[\mathrm{0,1}\right]}{\max }\mathcal{J}\left(\gamma \left(t\right)\right),$$

(19)

where

$$\mathsf{\Gamma }=\left\{\gamma \in \left.\complement \left(\left[\mathrm{0,1}\right],H^1\left({\mathbb{R}}^{\mathrm{N}}\right)\right)\right| \gamma \left(0\right)=0, \mathcal{J}\left(\gamma \left(1\right)\right)<0\right\}.$$

Theorem 2. 

We assume that $\alpha \in \left(0,N\right)$, $N\ge 3$ and $k_{*}=\frac{\alpha +2}{2\left(N+\alpha \right)}{\aleph }^{\frac{N+\alpha }{N+2}}.$ Then, $k_0<k_{*}.$

Proof. 

We firstly show that $k_0\le k_1,$ where

$$k_1=\underset{u\in H^1\left({\mathbb{R}}^{\mathrm{N}}\right)\backslash \left\{0\right\}}{\mathit{inf}}\underset{t\ge 0}{\max }\mathcal{J}\left(tu\right),$$

Indeed, for any $u\in H^1\left({\mathbb{R}}^{\mathrm{N}}\right)\backslash \left\{0\right\},$ $t_u>0$ exists according to Theorem 1 such that $\mathcal{J}\left(t_{u}u\right)<0.$ As a result, according to the definition of $k_0$, we have

$$k_0\le \underset{\tau \in \left[\mathrm{0,1}\right]}{\max }\mathcal{J}\left(\tau t_{u}u\right)\le \underset{t\ge 0}{\max }\mathcal{J}\left(tu\right).$$

(20)

Accordingly, we have $k_0\le k_1$ given that the lefthand side does not depend on the choice of $u.$ By applying a representation Formula (14) for the optimal functions of the Hardy$-$Littlewood$-$Sobolev inequality, we set the following for $\epsilon >0$ and for each $x\in {\mathbb{R}}^{\mathrm{N}}:$

$$u_{\epsilon }\left(x\right)=A{\left(\frac{{\epsilon }^{\frac{1}{2}}}{\epsilon +{\left|x\right|}^2}\right)}^{\frac{N-2}{2}} \mathrm{and} u_1\left(x\right)=A{\left(\frac{1}{1+{\left|x\right|}^2}\right)}^{\frac{N-2}{2}}.$$

For each $\epsilon >0,$ the function, let $\mathfrak{R}\in {\complement }_0^{\infty }({\mathbb{R}}^{\mathrm{N}},\left[\mathrm{0,1}\right])$ be a cut-off function satisfying

$$\mathfrak{R}=\left\{\begin{array}{c}1, x\in B_{\rho },\\ 0, x\in {\mathbb{R}}^{\mathrm{N}}\backslash B_{2\rho ,}\end{array}\right.$$

where $\rho$ is some positive constant. Set $U_{\epsilon }=\mathfrak{R}{u}_{\epsilon }$ satisfies

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|\nabla U_{\epsilon }\right|}^2=\aleph ,$$

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|U_{\epsilon }\right|}^{\frac{N+\alpha }{N-2}}\right){\left|U_{\epsilon }\right|}^{\frac{N+\alpha }{N-2}}=1.$$

Now, we consider a function ${\mathfrak{B}}_{\epsilon }:[0,+\infty )\to \mathbb{R}$ for every $\epsilon >0$and for each $t\in \left[0,+\infty \right),$ and it is defined as

$${\mathfrak{B}}_{\epsilon }\left(t\right):=\mathcal{J}\left(t{U}_{\epsilon }\right)=\phi \left(t\right)+\psi \left(t\right),$$

where functions $\phi :[0,+\infty )\to \mathbb{R}$ and $\psi :[0,+\infty )\to \mathbb{R}$ for each $t\in [0,+\infty )$ are defined by

$$\phi \left(t\right)=\frac{1}{2}\aleph t^2-\frac{N-2}{2\left(N+\alpha \right)}{t}^{\frac{2\left(N+\alpha \right)}{N-2}}$$

and

$$\psi \left(t\right)=\frac{1}{2}{t}^2{\int }_{{\mathbb{R}}^{\mathrm{N}}}{V\left|U_{\epsilon }\right|}^2.$$

Given that ${\mathfrak{B}}_{\epsilon }\left(t\right)>0$ whenever $t>0$ is sufficiently small,

$$\underset{t\to 0}{\lim }{\mathfrak{B}}_{\epsilon }\left(t\right)=0$$

and

$$\underset{t\to +\infty }{\lim }{\mathfrak{B}}_{\epsilon }\left(t\right)=-\infty ,$$

For each $\epsilon >0$ there exist $t_{\epsilon }>0$ such that

$${\mathfrak{B}}_{\epsilon }\left(t_{\epsilon }\right)=\underset{t\ge 0}{\sup }{\mathfrak{B}}_{\epsilon }\left(t\right)=\underset{t\ge 0}{\max }{\mathfrak{B}}_{\epsilon }\left(t\right).$$

According to the definition of function $\phi ,$ we have

$$k_1\le \underset{t\ge 0}{\max }{\mathfrak{B}}_{\epsilon }\left(t\right)={\mathfrak{B}}_{\epsilon }\left(t_{\epsilon }\right)=\phi \left(t_{\epsilon }\right)+\psi \left(t_{\epsilon }\right)<\phi \left(t_{*}\right)+\psi \left(t_{\epsilon }\right),$$

(21)

where $t_{*}={\aleph }^{\frac{N-2}{2(\alpha +2)}}>0$ is unique and the following is achieved

$$\phi \left(t_{*}\right)=\underset{t\ge 0}{\max }\phi \left(t\right)=\frac{\alpha +2}{2(N+\alpha )}{\aleph }^{\frac{N+\alpha }{\alpha +2}}=k_{*}.$$

Considering that $\acute{{\mathfrak{B}}_{\epsilon }}\left(t_{\epsilon }\right)=0,$ we have

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}{V\left|U_{\epsilon }\right|}^2+\aleph ={t_{\epsilon }}^{\frac{N-2}{2\left(\alpha +2\right)}}.$$

We have $\underset{\epsilon \to 0}{\lim }\mathit{max}{t_{\epsilon }}^{\frac{N-2}{2\left(\alpha +2\right)}}\le \mathrm{\aleph },$ which is equivalent to $\underset{\epsilon \to 0}{\lim }\mathit{sup}{t}_{\epsilon }<t_{\mathrm{*}}.$

We have $\underset{\epsilon \to 0}{\lim }{t_{\epsilon }}^{\frac{N-2}{2\left(\alpha +2\right)}}=\mathrm{\aleph }$ and thus $\underset{\epsilon \to 0}{\lim }{t}_{\epsilon }=t_{\mathrm{*}}.$ We now observe that

$$\psi \left(t_{\epsilon }\right)=\frac{{t_{\epsilon }}^2}{2}{\int }_{{\mathbb{R}}^{\mathrm{N}}}V{\left|U_{\epsilon }\right|}^2.$$

By changing the variables,

$$\psi \left(t_{\epsilon }\right)=\frac{{t_{\epsilon }}^2}{2}{\int }_{{\mathbb{R}}^{\mathrm{N}}}V(\frac{y}{\epsilon })\frac{A}{1+{\left|y\right|}^2}dy.$$

Using Lebesgue’s theorem of dominated convergence

$$\underset{\epsilon \to 0}{\lim }sup{\int }_{{\mathbb{R}}^{\mathrm{N}}}V\left(\frac{y}{\epsilon }\right)\frac{A}{1+{\left|y\right|}^2}dy\le {\int }_{{\mathbb{R}}^{\mathrm{N}}}\frac{A}{1+{\left|y\right|}^2}dy=\aleph .$$

According to (21), $k_1<k_{*}$; thus, $k_0<k_{*}$ in view of (20). □

Theorem 3. 

If ${\left(u_n\right)}_{n\in \mathbb{N}}$ is the sequence content in $H^1\left({\mathbb{R}}^N\right)$ such that

$$\underset{n\to \infty }{\lim }\mathit{inf}‖u_n‖>0, \underset{n\to \infty }{\lim }〈\acute{\psi }\left(u_n\right),u_n〉=0,$$

where a functional $\psi : H^1\left({\mathbb{R}}^{\mathrm{N}}\right)⟶\mathbb{R}$ is defined as follows:

$$\psi \left(u\right)=\frac{1}{2}{‖u‖}^2-\frac{N-2}{2(N+\alpha )}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}},$$

then $\underset{n\to \mathrm{\infty }}{\lim }\mathit{inf}\psi \left(u_n\right)\ge k_{\mathrm{*}}.$

Proof. 

We notice that, as $n\to \infty$

$${‖u_n‖}^2={\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}.$$

We deduce by the assumption $\underset{n\to \infty }{\lim }\mathit{inf}‖u_n‖>0$ and by the Hardy$-\mathrm{L}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{w}\mathrm{o}\mathrm{o}\mathrm{d}–$Sobolev inequality (11), that

$$\underset{n\to \infty }{\lim }\mathit{inf}{\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|u_n\right|}^2>0.$$

From the definition of $\aleph$and given that $n\to \infty ,$ we have

$$\begin{array}{c}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\ge {\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|u_n\right|}^2 \\ \ge \aleph {\left({\int }_{{\mathbb{R}}^N}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{N-2}{N+\alpha }}\end{array}$$

which leads to

$$\underset{n\to \infty }{\lim }\mathit{inf}{‖u_n‖}^2=\underset{n\to \infty }{\lim }inf{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\ge {\aleph }^{\frac{N+\alpha }{\alpha +2}}.$$

(22)

Therefore,

$$\psi \left(u_n\right)=\psi \left(u_n\right)-\frac{N-2}{2(N+\alpha )}〈\acute{\psi }\left(u_n\right),u_n〉$$

$$=\frac{\alpha +2}{2(N+\alpha )}{‖u_n‖}^2.$$

(23)

Then, the conclusion follows from (22) and (23). □

Even though the functional $\mathcal{J}$ does not have global satisfaction of the Palais–Smale condition, it remains applicable within a defined energy level.

Theorem 4. 

For the function $\mathcal{J},$ with the assumption that ${\left(u_n\right)}_{n\in \mathbb{N}}$ is the bounded sequence and satisfies (PS)_k sequence with  $k\in \left(0,k_{*}\right).$Then, up to the subsequence and translations, the sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ converges weakly to some functions $u\in H^1\left({\mathbb{R}}^N\right)\backslash \left\{0\right\}$ such that

$$\mathcal{J}\left(u\right)\in \left(0, k\right] and \acute{\mathcal{J}}\left(u\right)=0.$$

Proof. 

Firstly, we prove that

$$\underset{n\to \infty }{\lim }\sup {\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|u_n\right|}^q>0.$$

Given $\underset{n\to \mathrm{\infty }}{\lim }〈\acute{\mathcal{J}}\left(u_n\right),u_n〉=0,$ that

$${‖u_n‖}^2={\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}.$$

On the other hand, $\mathcal{J}\left(u_n\right)\to k>0$ as $n\to \infty ,$ which combined with the Hardy$-$Littlewood$-$Sobolev inequality (11), implies that

$$\underset{n\to \infty }{\lim }\inf ‖u_n‖>0.$$

We, thus, deduce from Theorem 3 that

$$k\ge \underset{n\to \infty }{\lim }\inf \mathcal{J}\left(u_n\right)=\underset{n\to \infty }{\lim }\inf \psi \left(u_n\right)\ge k_{*},$$

This condition would be a contradiction.

By using the Lions inequality ([38], Lemma 1.21), ([39], Lemma 1.1) and ([40], (2.4)), we have

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|u_n\right|}^q\le C\left({\int }_{{\mathbb{R}}^N}\left({\left|\nabla u_n\right|}^2+V\left(x\right){\left|u_n\right|}^2\right)\right) {\left(\underset{y\in {\mathbb{R}}^N}{\mathit{sup}}{\int }_{B_1\left(y\right)}{\left|u_n\right|}^q\right)}^{1-\frac{2}{q}},$$

Consequently, the sequence of points ${\left(y_n\right)}_{n\in \mathbb{N}}$ in ${\mathbb{R}}^{\mathrm{N}}$ exists such that

$$\underset{n\to \infty }{\lim }\inf {\int }_{B_1\left(y\right)}{\left|u_n\right|}^q>0.$$

We then define $\check{u_n}:=u_n\left(.+y_n\right)$ because the function $\mathcal{J}$ is invariant under translation. The sequence ${\left(\check{u_n}\right)}_{n\in \mathbb{N}}\subset {\mathsf{H}}^1\left({\mathbb{R}}^N\right)$ is also a bounded (PS)_k sequence that converges weakly to some functions $u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right)\backslash \left\{0\right\}.$

We now claim that $\acute{\mathcal{J}}\left(u\right)=0.$ For simplicity, the sequence ${\left(\check{u_n}\right)}_{n\in \mathbb{N}}$ is still denoted by ${\left(u_n\right)}_{n\in \mathbb{N}}.$ Given that

$${ u}_n\stackrel{weakly}{\to } u, in {\mathsf{H}}^1\left({\mathbb{R}}^N\right),$$

through the application of the Sobolev$-$Rellich embedding theorem,

$$u_n\stackrel{strongly }{\to } u, in { L}_{loc}^2\left({\mathbb{R}}^N\right),$$

and still, up to the subsequence,

$$u_n⟶ u, in \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{n} {\mathbb{R}}^N.$$

Given that the sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is bounded in $L^2\left({\mathbb{R}}^N\right),$ the sequence ${\left({\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right)}_{n\in \mathbb{N}}$ is bounded and, thus, converges weakly to ${\left|u\right|}^{\frac{N+\alpha }{N-2}}$ in $L^{\frac{2N}{N+\alpha }}\left({\mathbb{R}}^N\right)$ ([41], Proposition 5.4.7).

$${\left|u_n\right|}^{\frac{N+\alpha }{N-2}} \stackrel{weakly}{\to }{ \left|u\right|}^{\frac{N+\alpha }{N-2}}, \mathrm{in} L^{\frac{2N}{N+\alpha }}\left({\mathbb{R}}^N\right).$$

Given that a Riesz potential is a linear bounded map from $L^{\frac{2N}{N+\alpha }}$ to $L^{\frac{2N}{N-\alpha }}\left({\mathbb{R}}^N\right)$, we have

$${\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right)}_{n\in \mathbb{N}}\stackrel{weakly}{\to } \left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right), in L^{\frac{2N}{N-\alpha }}\left({\mathbb{R}}^N\right).$$

Consequently, for any $\varphi \in {\complement }_k^{\infty }\left({\mathbb{R}}^N\right)$ and given that $n\to \infty ,$

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}{u}_n\varphi \to {\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}u\varphi .$$

which is together with a fact that a smooth test function set ${\complement }_k^{\infty }\left({\mathbb{R}}^N\right)$ is dense in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right)$ given that $\acute{\mathcal{J}}(u)=0.$

We consider $\tilde{\mathfrak{I}}=min\left\{2\left(\frac{N+\alpha }{N-2}\right),\mathfrak{I}\right\}>2,$ Consequently, based on Fatou’s lemma, we conclude that

$$\begin{array}{c}\mathcal{J}\left(u\right)=\mathcal{J}\left(u\right)-\frac{1}{\mathfrak{I}}〈\acute{\mathcal{J}}\left(u\right),u〉 \\ =\left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖u‖}^2+\left(\frac{1}{\mathfrak{I}}-\frac{N-2}{2(N+\alpha )}\right){\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\\ \le \left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖u_n‖}^2+\left(\frac{1}{\mathfrak{I}}-\frac{N-2}{2(N+\alpha )}\right){\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\\ =\mathcal{J}\left(u_n\right)-\frac{1}{\mathfrak{I}}〈\acute{\mathcal{J}}\left(u_n\right),u_n〉 \to k. \end{array}$$

In the end, we conclude that

$$\mathcal{J}\left(u\right)=\mathcal{J}\left(u\right)-\frac{1}{\mathfrak{I}}〈\acute{\mathcal{J}}\left(u\right),u〉\ge \left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖u‖}^2>0.\square$$

Theorem 5. 

For each$N\ge 3$and$\alpha \in \left(0,N\right)$. Then, the Choquard Equation (6) has a ground state solution.

Proof. 

Two steps will be used to prove this theorem. Firstly, we find the nontrivial solution to problem (6) with an energy level that is strictly less than $k_{*}.$ Thereafter, we present that the minimization problem

$${\delta }_0=inf\left\{\left.\mathcal{J}\left(u\right)\right| u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right)\backslash \left\{0\right\} and \acute{\mathcal{J}}\left(u\right)=0\right\},$$

(24)

is attained.

By using Theorem 1 and a mountain pass theorem, the Palais$-$Smale sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is proven to exist at the energy level $k_0$ defined by (19). Then, Theorem 2 indicates that $k_0\in \left(0,k_{*}\right).$ The sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is a bounded in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right)$; actually, by taking $\tilde{\mathfrak{I}}=min\left\{2\left(\frac{N+\alpha }{N-2}\right),\mathfrak{I}\right\}>2,$ we find that

$$\begin{array}{c}{k}_0+‖u_n‖\ge \mathcal{J}\left(u_n\right)-\frac{1}{\mathfrak{I}}〈\acute{\mathcal{J}}\left(u_n\right),u_n〉 \\ =\frac{1}{2}{‖u_n‖}^2-\frac{N-2}{2(N+\alpha )}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-∆\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\\ -\left[\frac{1}{\mathfrak{I}}{‖u_n‖}^2-\frac{1}{\mathfrak{I}}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-∆\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right]\\ =\left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖u_n‖}^2+\left(\frac{1}{\mathfrak{I}}-\frac{N-2}{2(N+\alpha )}\right){\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\\ \ge \left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖u_n‖}^2. \end{array}$$

As a result, up to the subsequence and given that $n\to \infty$

$$u_n\stackrel{weakly}{\to } u \mathrm{in} {\mathsf{H}}^1\left({\mathbb{R}}^N\right).$$

Through a classical Sobolev$-$Rellich embedding theorem, we can deduce that

$${ u}_n\stackrel{strongly }{\to } u, in L_{loc}^q\left({\mathbb{R}}^N\right),$$

and $u_n$ converges to $u$ nearly everywhere in ${\mathbb{R}}^N.$ According to Theorem 4, we deduce that $u$ is a nontrivial critical point of a functional $\mathcal{J}$ and $\mathcal{J}\left(u\right)\in \left(0,k_0\right].$

We now show the existence of a minimizer for the minimization issue stated by (24). We assume that ${\left(v_n\right)}_{n\in \mathbb{N}}$ is the sequence of nontrivial solutions to (6) such that $\underset{n\to \infty }{\lim }\mathcal{J}\left(v_n\right)={\delta }_0.$ We firstly observe that ${\delta }_0\le k_0<k_{*}.$ Given that $\acute{\mathcal{J}}\left(v_n\right)=0$ and by taking $\tilde{\mathfrak{I}}=min\left\{2\left(\frac{N+\alpha }{N-2}\right),\mathfrak{I}\right\}>2,$ we have

$${ \delta }_0=\mathcal{J}\left(v_n\right)-\frac{1}{\mathfrak{I}}〈\acute{\mathcal{J}}\left(v_n\right),v_n〉$$

$$\ge \left(\frac{1}{2}-\frac{1}{\mathfrak{I}}\right){‖v_n‖}^2.$$

(25)

Consequently, a sequence ${\left(v_n\right)}_{n\in \mathbb{N}}$ is a bounded in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right).$

$${‖v_n‖}^2={\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|v_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|v_n\right|}^{\frac{N+\alpha }{N-2}}$$

$$\le C{\left({\int }_{{\mathbb{R}}^{\mathrm{N}}}{\left|v_n\right|}^2\right)}^{\frac{N+\alpha }{N-2}}$$

$$\le C{‖v_n‖}^{2\left(\frac{N+\alpha }{N-2}\right)}.$$

As a result, $\underset{n\to \infty }{\lim }‖v_n‖>0.$ It implies that ${\delta }_0$ when it is combined with (25)$.$ The sequence ${\left(v_n\right)}_{n\in \mathbb{N}}$ is a bounded ${\left(\mathrm{P}\mathrm{S}\right) }_{{\delta }_0}$ sequence for a functional $\mathcal{J}$. Thus, we infer from Theorem 4 that, up to the subsequence and translations,

$$v_n\stackrel{weakly}{\to }v \mathrm{in} {\mathsf{H}}^1\left({\mathbb{R}}^N\right) \mathrm{as} n\to \infty ,v\ne 0$$

and

$$\acute{\mathcal{J}}\left(v\right)=0 and \mathcal{J}\left(v\right)\in \left(0,{\delta }_0\right].$$

In the meantime, we establish that $\mathcal{J}\left(v\right)={\delta }_0$based on the definition of ${\delta }_0.$ Consequently, $v$ is the ground state solution of (6). □

Theorem 6. 

We consider that $V\left(x\right)\in L^{\infty }\left({\mathbb{R}}^N\right)$ and

$$\underset{\left|x\right|\to \infty }{\mathit{lim}}infV\left(x\right)\ge 1.$$

(26)

If ${\aleph }_{*}<\aleph$, then the infimum ${\aleph }_{*}$ is attained. In ${\mathsf{H}}^1\left({\mathbb{R}}^N\right),$ every minimising sequence for ${\aleph }_{*}$ up to the subsequence converges strongly.

Proof. 

To classify the existence of nontrivial solutions for a Choquard Equation (6), the critical level is defined as follows

$${\aleph }_{*}=inf\left\{\left.{\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2+V{\left|u\right|}^2\right|u{\in \mathsf{H}}^1\left({\mathbb{R}}^N\right) and {\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}=1\right\}.$$

(27)

Let ${\left(u_n\right)}_{n\in \mathbb{N}}$ $\subset {\mathsf{H}}^1\left({\mathbb{R}}^N\right)$be a minimizing sequence for ${\aleph }_{*},$that is,

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}=1,$$

$$\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2+V{\left|u_n\right|}^2\to {\aleph }_{*}.$$

By using (26), we can find that the sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ is bounded in ${\mathsf{H}}^1\left({\mathbb{R}}^N\right).$ As a result, $u\in {\mathsf{H}}^1\left({\mathbb{R}}^N\right)$ exists such that, up to the subsequence,

$$u_n\stackrel{weakly}{\to } u, in {\mathsf{H}}^1\left({\mathbb{R}}^N\right),$$

and, by a classical Rellich$-$Kondrachov compactness theorem,

$$u_n\stackrel{strongly }{\to } u, in { L}_{loc}^2\left({\mathbb{R}}^N\right),$$

by the lower semi-continuity of the norm under weak convergence,

$${\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2+V{\left|u\right|}^2\le \underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2+V{\left|u_n\right|}^2={\aleph }_{*}$$

and by Fatou’s lemma

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n\right|}^{\frac{N+\alpha }{N-2}}\le 1.$$

In conclusion, it suffices to show that equality is attained in a latter inequality. By using Brezis$-$Lieb lemma for the Riesz potential ([10], Lemma 2.4), we note that

$$\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}=1-{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}.$$

(28)

while by Lemma 2 and by the lower semicontinuity of the norm under weak convergence, we note that

$${\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2+V{\left|u\right|}^2\le \underset{n\to \infty }{\lim }\mathit{inf}{\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2+V{\left|u_n\right|}^2+\underset{n\to \infty }{\lim }\mathit{inf}{\int }_{{\mathbb{R}}^N}V{\left|u_n\right|}^2-{\left|u_n-u\right|}^2$$

$$\le \underset{n\to \infty }{\lim }\mathit{inf}{\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2+V{\left|u_n\right|}^2-{\left|u_n-u\right|}^2$$

$$\le {\aleph }_{*}+\underset{n\to \infty }{\lim }\mathit{sup}{\int }_{{\mathbb{R}}^N}{\left|u_n-u\right|}^2.$$

(29)

By definition of $\aleph$, we have

$${\int }_{{\mathbb{R}}^N}{\left|u_n-u\right|}^2\ge \aleph {\left({\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u_n-u\right|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{N-2}{N+\alpha }}.$$

(30)

Therefore, we conclude that

$${\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2+V{\left|u\right|}^2\le {\aleph }_{\ast }-\aleph {\left(1-{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{N-2}{N+\alpha }}.$$

In view of a definition of ${\aleph }_{*},$ this condition implies that

$${\aleph }_{*}\ge \aleph {\left(1-{\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{N-2}{N+\alpha }}+{\aleph }_{*}{\left({\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{N-2}{N+\alpha }}.$$

From the assumption ${\aleph }_{*}<\aleph ,$ we conclude that

$${\int }_{{\mathbb{R}}^{\mathrm{N}}}\left({\left(-{∆}^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}\right){\left|u\right|}^{\frac{N+\alpha }{N-2}}=1.$$

And, hence, according to the definition of ${\aleph }_{*}$, we have

$${\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2+V{\left|u\right|}^2={\aleph }_{*}.$$

That is, the infimum ${\aleph }_{*}$ is attained at $u$. From (26), we establish that $u_n\to u$ in $L^2\left({\mathbb{R}}^N\right)$. Given that $V\left(x\right)\in L^{\infty }\left({\mathbb{R}}^N\right),$ we obtain that ${Vu}_n\to Vu$ in $L^2\left({\mathbb{R}}^N\right)$. By using (26) again, we conclude that

$${\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2=\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^N}{\left|{\nabla u}_n\right|}^2.$$

Since

$$u_n\stackrel{weakly}{\to } u, in {\mathsf{H}}^1\left({\mathbb{R}}^N\right),$$

it follows that

$$u_n\stackrel{strongly }{\to } u, in { L}_{loc}^2\left({\mathbb{R}}^N\right).\square$$

Remark 1 

([42]). In (6), if $\alpha =0$ ,then we obtain

$$-∆u+V\left(x\right)u={\left|u\right|}^{\frac{2\alpha +4}{N-2}}u.$$

In (6), if $V=1$, then the problem has no solution [10].

4. Conclusions

In this article, we focused on a fundamental problem: establishing the existence of a ground state solution for an upper critical Choquard equation, a challenge that holds significance in the fields of mathematics and physics. We achieved several notable outcomes, which contribute to our understanding of this complex equation and its practical applications. Our research successfully confirmed the existence of a ground state solution for the upper critical Choquard equation. This outcome addresses a long-standing question and extends our knowledge of nonlinear partial differential equations. By considering a Riesz potential represented as ${\left(-{∆}^D\right)}^{\frac{-\mathit{\alpha }}{2}}$, where ${∆}^D$ is a Laplacian operator on $\mathsf{\Omega }$ with Dirichlet boundary conditions, we added depth to our understanding of the behavior of Choquard equations under these specific conditions. Our work demonstrated that the existence of a ground state solution is not limited to abstract theoretical contexts. We established this solution within bounded domains $\mathsf{\Omega }\subset {\mathbb{R}}^N$, which is particularly relevant in physical and mathematical analysis involving Choquard equations. Variational methods proved to be instrumental in our research, enabling us to address the problem effectively. This underscores the importance of such techniques in handling intricate nonlinear phenomena.

In conclusion, our study has not only resolved the question of ground state existence for upper critical Choquard equations but has also enhanced our comprehension of the role of Riesz potentials and Laplacians with Dirichlet boundary conditions in this context. The practical applicability of our findings in bounded domains further underlines the relevance of this research. We anticipate that our work will inspire future investigations in the field of nonlinear partial differential equations, contributing to the broader understanding of the Choquard equation and its associated phenomena.