Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

In this paper, we focus on the existence of solutions for the Choquard equation {−Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+λ|u|p−2u,x∈RN;u∈H1(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ \end{document} where λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >0$\end{document} is a parameter, α∈(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in (0,N)$\end{document}, N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\ge 3$\end{document}, Iα:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$\end{document} is the Riesz potential. As usual, α/N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha /N+1$\end{document} is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if λ>λ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >\lambda _{*}$\end{document} for some given number λ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{*}$\end{document} in three cases: (i) 2<p<4N+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2< p<\frac{4}{N}+2$\end{document}, (ii) p=4N+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p=\frac{4}{N}+2$\end{document}, and (iii) 4N+2<p<2∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{4}{N}+2< p<2^{*}$\end{document}. Our result improves the previous related ones in the literature.

If f = 0 and V (x) ≡ 1, (1.9) appears under the background of various physical models. For example, as early as in 1954, Pekar [2] introduced (1.9) into the physical model to study the free electrons in a ionic lattice interact with phonons associated with deformations of the lattice. Choquard equation is also known as the Schrödinger-Newton equation after the addition of non-relativistic Newtonian gravity to some Schrödinger equations [3][4][5][6]. Lieb [7] first verified the positive solution of (1.9) in R 3 when f = 0, α = 2, V (x) ≡ 1, and q = 2. Later, Lions [8,9] further improved the results of (1.9) and obtained the existence and multiplicity of normalized solution for (1.9). The existence of a ground state solution and the qualitative properties of the solution in the range of exponents q which satisfies were established in [10]. The endpoints N+α N-2 and N+α N are critical exponents. It is known to all that N+α N-2 is an upper critical exponent which plays a similar role as the Sobolev critical exponent in the local semilinear equations [11][12][13][14][15][16][17]. The lower critical exponent N+α N is strictly greater than 1 which comes from inequality (1.2). So far, many authors have investigated the existence of nontrivial solutions of many forms of (1.9) (see [18][19][20][21]). In addition, for some applications of the variational method in elliptic systems, we refer to [22][23][24]. If the potential V (x) ≡ 1, then (1.1) reduces to the following equation: (1.10) Tang, Wei, and Chen [25] proved that (1.10) has ground state solutions in the following assumptions: (i) 2 < p < 4 N + 2 and λ > 0; By using the mountain pass lemma, they obtained a Palais-Smale sequence and the corresponding energy level m. Then, from these three assumptions, an estimate of the energy level m was given, which is very important to ensure the Sobolev compactness. We further improve these three hypotheses to be applicable to the research in this paper. This has certain enlightenment to our work.
Motivated by the work of [26,27], we use a weaker decay assumption on ∇V to solve the trouble caused by variable potential.
(V3) V ∈ C 1 (R N , R), and there is θ ∈ [0, 1) such that Van Schaftingen and Xia [11], Chen and Tang [26] did a pretty good job, which gives us some inspiration. To our knowledge, there seems to be no results of (1.1). Motivated by the above works, especially [25,26], in this paper, we establish the existence result of ground state solutions for (1.1). To state our result, inspired by [28], we define the following Pohožaev identity functional on H 1 (R N ): and In view of [29, Prorosition 3.1], ifū is a solution of (1.1), then it satisfies the Pohožaev identity P(u) = 0. Let Our main result is as follows.
Theorem 1.1 Assume that V satisfies (V1)-(V3) and one of the following conditions: holds. Then problem (1.1) has a solutionū ∈ H 1 (R N ) such that where u t (x) := u(x/t).
In this paper, we use the following notations: • H 1 (R N ) denotes the usual Sobolev space equipped with the inner product and the • L s (R N ) (1 < s < ∞) denotes the Lebesgue space with the norm u s = ( R N |u| s dx) 1/s . • For any u ∈ H 1 (R N ) and r > 0, B r (x) := {y ∈ R : |y -x| < r}.

Proof of the main result
Before proving the main result, we first give some key inequalities and lemmas. The following famous Hardy-Littlewood-Sobolev inequality [1,Theorem 4.3] is an origin of the variational approach to (1.1).
By a simple calculation, we have the following lemma.

Lemma 2.2
The following two inequalities hold: Moreover, (V3) implies that the following inequality holds:

Lemma 2.3 Assume that (V1) and (V3) hold. Then
Proof According to Hardy's inequality, we obtain Note that From Lemma 2.3, we have the following corollary.

Corollary 2.4
Assume that (V1) and (V3) hold. Then, for u ∈ M, Based on the above results, we establish the following important property for M.
Not unnaturally, we claim that t u is unique for any u ∈ H 1 (R N ) \ {0}. As a matter of fact, for any given u ∈ H 1 (R N ) \ {0}, if there are two positive constants t 1 = t 2 such that u t 1 , u t 2 ∈ M, then P(u t 1 ) = P(u t 2 ) = 0. Together with (2.3), (2.4), and (2.5), we have (2.10) The same procedure may be easily adapted to obtain the following equation: From (2.10) and (2.11), we have u t 1 = u t 2 , which shows that t u > 0 is unique for any Then there are two constants γ 1 , γ 2 > 0 such that Proof The proof of Lemma 2.6 is routine, and we omit it.
From Corollary 2.4 and Lemma 2.5, we have M = ∅. Next, we apply the method introduced in [26] to prove the following lemma, which is key to verifying the minimax characterization.
Inspired by Tang and Chen [25], we give an estimate on the energy level m, which is essential in ensuring compactness.
Proof We set U(x) = A 0 (1 + |x| 2 ) -N 2 , where A 0 is defined by (1.4). By the calculation of integral, we get and For any ε > 0, we define two functions f (t) and h ε (t) as follows:      . In this case we choose ε ∈ (0, 1), then (2.43) By assumption (ii) in Theorem 1.1, we can choose > 0 such that We choose ε > 0 such that . In this case, we also choose ε ∈ (0, 1], then There are four possible subcases. Subcase (i) t ≥ T 2 . Then it follows from (2.29), (2.33), (2.51), and (2.50) that The above three cases show that This is to show that { ∇u n 2 } is bounded. Next, we prove that { u n 2 } is also bounded. Arguing indirectly, assume that u n 2 → ∞, without loss of generality, we can assume that u n 2 ≥ 1. From (2.28), we have Passing to a subsequence, we have u n ū in H 1 (R N ). Then u n →ū in L s loc (R N ) for 2 ≤ s ≤ 2 * and u n →ū a.e. in R N . We obtain two possible cases.