Existence Results for Fractional Choquard Equations with Critical or Supercritical Growth

In this paper, we study the following fractional Choquard equation with critical or supercritical growth (-Δ)su+V(x)u=f(x,u)+λ|x|-μ∗|u|pp|u|p-2u,x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ (-\Delta )^su+V(x)u=f(x,u)+\lambda \left[ |x|^{-\mu }*|u|^p\right] p|u|^{p-2}u, \quad x \in {\mathbb {R}}^N, \end{aligned}$$\end{document}where 0<s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1$$\end{document}, (-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} denotes the fractional Laplacian of order s, N>2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>2s$$\end{document}, 0<μ<2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\mu <2s$$\end{document} and p≥2μ,s∗:=2N-μN-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 2_{\mu ,s}^*:=\frac{2N-\mu }{N-2s}$$\end{document}, which is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. Under some suitable conditions, we prove that the equation admits a nontrivial solution for small 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} by variational methods, which extends results in Bhattarai in J. Differ. Equ. 263, 3197–3229 (2017).


Introduction and main result
Consider the following fractional Choquard equation where 0 < s < 1 , (−Δ) s denotes the fractional Laplacian of order s, N > 2s , 0 < < 2s and p ≥ 2 * ,s ∶= 2N− N−2s . Problem (1.1) has nonlocal characteristics in the nonlinearity as well as in the (fractional) diffusion. When s = 1, = 1, = 1, p = 2 and f (x, u) = 0 , then (1.1) boils down to the so-called Choquard equation which goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954 [15] and the modeling of an electron trapped in its own hole in 1976 in the work of Choquard, as a certain approximation to Hartree-Fock theory of onecomponent plasma [6]. In some particular cases, this equation is also known as the Schrödinger-Newton equation, which was introduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanical wave function [16]. The first investigations for existence and symmetry of the solutions to (1.2) go back to the works of Lieb [6] and Lions [7]. Since then many efforts have been made to study the existence of nontrivial solutions for nonlinear Choquard equations, see for instance [3,12,13].
For fractional Laplacian with nonlocal Hartree-type nonlinearities, the problem has also attracted a lot of interest, we refer to Refs. [2,4,5,8,10,11] and their references therein.
Most of the works afore mentioned are set in ℝ N , N > 2s , with subcritical and critical growth nonlinearities and to the authors' best knowledge no results are available on the existence for problem (1.1) with supercritical exponent. We aim at studying the existence of nontrivial solutions for critical or supercritical problem (1.1).
In order to reduce the statements for main result, we list the assumption as follows: For any 0 < s < 1 , the fractional Sobolev space H s ℝ N is defined by endowed with the natural norm where the term Our main result is the following:  [9] (Hardy-Littlewood-Sobolev inequality) Let r, t > 1 and

Remark 1.2 Bhattarai in [1] studied the following fractional Schrödinger equation
Then there exists a sharp constant C r,N, ,t independent of g and h such that It is well known to us that a weak solution of problem (1.1) is a critical point of the following functional Clearly, we cannot apply variational methods directly because the functional I is not well defined on E unless p = 2 * ,s . To overcome this difficulty, we define a function t 0 (s)ds ≥ 0 and | (t)| ≤ pM p−q |t| q−1 for all t ∈ ℝ . Moreover, there exists a constant C > 0 such that for all u ∈ H s ℝ N . Indeed, for any u ∈ H s ℝ N , taking t ∈ N N− , 2N q (N−2s) , by the Hölder inequality we can calculate that

Let
By mountain pass theorem, using a standing argument we can prove that the equation Journal of Nonlinear Mathematical Physics (2022) 29:859-868 We can use a similar argument to obtain the above conclusion if a ≤ b . Therefore, for all a, b ∈ ℝ . Consequently, which implies that By the fact that Γ(u ) ≥ 1 u u −1 ,L we see that where S * = S(N, s) > 0 is a sharp constant that satisfies S * ‖u‖ 2 2 * s ≤ [u] 2 for any u ∈ H s (ℝ N ) ( [14]). By the proof of (2.1) we know that there exists a constant C 0 > 0 such that Moreover, by virtue of (f 1 )-(f 2 ) we know that for any > 0 , there exists C > 0 such that for all (x, t) ∈ ℝ N × ℝ . For fixed > 0 and small > 0 , by (2.5) and properties of we have for all (x, t) ∈ ℝ N × ℝ . Therefore, in view of (2.2)-(2.4) and (2.6) one has ,L (y) , Iterating this process and recalling that * s = 2 * s , we can infer that for every m ∈ ℕ, Let m → +∞ and recalling that ‖u ‖ 2 * s ≤ K we obtain This completes the proof. Availability of data and material Not applicable.

Conflict of interest
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