Existence of nontrivial solutions for fractional Schrödinger equations with electromagnetic fields and critical or supercritical nonlinearity

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: (−Δ)Asu+V(x)u=f(x,|u|2)u+λ|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}$$ (-\Delta )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert ^{2}\bigr)u+\lambda \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, $$\end{document} where (−Δ)As\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-\Delta )_{A}^{s}$\end{document} is the fractional magnetic operator with 02s\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}, 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}, 2s∗=2NN−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2_{s}^{*}=\frac{2N}{N-2s}$\end{document}, p≥2s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq 2_{s}^{*}$\end{document}, f is a subcritical nonlinearity, and V∈C(RN,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V \in C(\mathbb{R}^{N},\mathbb{R})$\end{document} and A∈C(RN,RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A \in C(\mathbb{R}^{N}, \mathbb{R}^{N})$\end{document} are the electric and magnetic potentials, respectively. Under some suitable conditions, by variational methods we prove that the equation has 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}. Our main contribution is related to the fact that we are able to deal with the case p>2s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p>2_{s}^{*}$\end{document}.


Introduction and preliminaries
Consider the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: where (-) s A is the fractional magnetic operator with 0 < s < 1, N > 2s, λ > 0, 2 * s = 2N N-2s , p ≥ 2 * s , f is a subcritical nonlinearity, and V ∈ C(R N , R) and A ∈ C(R N , R N ) are the electric and magnetic potentials, respectively.
The fractional magnetic Laplacian is defined by u(x)e i(x-y)·A( x+y 2 ) u(y) |x -y| N+2s dy, C N,s = 4 s Γ ( N+2s 2 ) π N 2 |Γ (-s)| . © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. This nonlocal operator has been defined in [4] as a fractional extension (for any s ∈ (0, 1)) of the magnetic pseudorelativistic operator or Weyl pseudodifferential operator defined with midpoint prescription [1]. As stated in [17], up to correcting the operator by the factor (1s), it follows that (-) s A u converges to -(∇u -iA) 2 u as s → 1. Thus, up to normalization, the nonlocal case can be seen as an approximation of the local one. The motivation for its introduction is described in more detail in [4,17] and relies essentially on the Lévy-Khintchine formula for the generator of a general Lévy process.
The main driving force for the study of problem (1.1) arises in the following timedependent Schrödinger equation when s = 1: where is the Planck constant, m is the particle mass, A : R N → R N is the magnetic potential, P : R N → R N is the electric potential, ρ is the nonlinear coupling, and ψ is the wave function representing the state of the particle. This equation arises in quantum mechanics and describes the dynamics of the particle in a nonrelativistic setting [2,15]. Clearly, the form ψ(x, t) := e -i th -1 u(x) is a standing wave solution of (1.2) if and only if u(x) satisfies the following stationary equation: where ε = , V (x) = 2m(P(x) -), and f = 2mρ; see [3,5,7,8]. By applying variational methods and Lyusternik-Schnirelmann theory Ambrosio and d' Avenia [1] proved the existence and multiplicity of solutions for the equation when ε > 0 is small. Recently, Liang et al. [14] obtained the existence and multiplicity of solutions for the fractional Schrödinger-Kirchhoff equation with the help of fractional version of the concentration compactness principle and variational methods. If the magnetic field A ≡ 0, then the operator (-) s A can be reduced to the fractional Laplacian operator (-) s , which is defined as The symbol P.V. stands for the Cauchy principal value, and C N,s is a dimensional constant that depends on N, s, precisely given by It is well known that the fractional Laplacian (-) s can be viewed as a pseudodifferential operator of symbol |ξ | 2s , as stated in Lemma 1.1 in [6]. Simultaneously, problem (1.1) be-comes the classical Schrödinger equation Recently, there has been a lot of interest in the study of equation (1.3) and other related nonlocal problems. See, for instance, [6, 10-13, 16, 21-23] and the references therein.
For more results about dealing with magnetic operators, see [9,20]. Nonlocal problems also appear in other mathematical research fields. We refer the interested readers to [18,19] for mathematical researches on Kirchhoff-type nonlocal equations, where Tang and Cheng [19] proposed a new approach to recover compactness for the (PS)-sequence, and Tang and Chen [18] proposed a new approach to recover compactness for the minimizing sequence.
Most of the works mentioned are set in R N , N > 2s, with subcritical or critical growth, and to the best of our knowledge, no results are available on the existence for problem (1.1) with supercritical exponent. In this paper, we aim at studying the existence of nontrivial solutions for critical or supercritical problem (1.1).
To reduce the statements of the main result, we introduce the following assumptions: Then we may introduce the Hilbert space and norm with the norm Define the norm on H s (R N ) as follows: Moreover, the best fractional critical Sobolev constant is given by Our main result is the following: with p ≥ 2 * s and μ > 0 has no nontrivial solution for all λ > 0. Indeed, let u ∈ E be a weak solution of the problem. Then we have the following Pohozaev identity: Moreover, taking u as the test function, we have Taking into account (1.4) and (1.5), we can derive that which implies the conclusion.

Proof of Theorem 1.1
It is well known 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 λ is not well defined on E unless p = 2 * s . To overcome this difficulty, we define the function where M > 0. Then φ ∈ C(R, R), φ(t)t ≥ qΦ(t) := q t 0 φ(τ ) dτ ≥ 0, and |φ(t)| ≤ M p-q |t| q-1 for all t ∈ R. Set h λ (x, t) = λφ(t) + f (x, |t| 2 )t for (x, t) ∈ R N × R. Then h λ (x, t) admits the following properties: By (h 1 )-(h 3 ), (V ), and the mountain pass theorem, using a standard argument, we easily see that the equation has a nontrivial solution u λ ∈ E with J λ (u λ ) = 0 and J λ (u λ ) = c λ := inf γ ∈Γ λ sup t∈[0,1] J λ (γ (t)), where We further set Then Γ ⊂ Γ λ and c λ ≤ c.

Lemma 2.1
The solution u λ satisfies u λ 2 ≤ 2q q-2 c λ , and there exists a constant A > 0 independent on λ such that u λ 2 ≤ A.