Abstract
In this paper, we study the following fractional Schrödinger–Poisson system with superlinear terms \t\t\t{(−Δ)su+V(x)u+K(x)ϕu=f(x,u),x∈R3,(−Δ)tϕ=K(x)u2,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta )^{s}u+V(x)u+K(x)\\phi u=f(x,u), & x \\in \\mathbb{R}^{3}, \\\\ (-\\Delta )^{t}\\phi =K(x)u^{2}, & x \\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where s,tin (0,1), 4s+2t>3. Under certain assumptions of external potential V(x), nonnegative density charge K(x) and superlinear term f(x,u), using the symmetric mountain pass theorem, we obtain the existence and multiplicity of non-trivial solutions.
Highlights
Introduction and main resultsIn this paper, we are concerned with the fractional Schrödinger–Poisson system ⎧⎨(– )su + V (x)u + K(x)φu = f (x, u), x ∈ R3,⎩(– )tφ = K (x)u2, x ∈ R3, (1.1)where (– )s is fractional Laplacian operator, s, t ∈ (0, 1), 4s + 2t > 3
We study the following fractional Schrödinger–Poisson system with superlinear terms
We are concerned with the fractional Schrödinger–Poisson system
Summary
Where (– )s is fractional Laplacian operator, s, t ∈ (0, 1), 4s + 2t > 3. On the potential V (x), we make the following assumptions: (V1) V (x) ∈ C(R3, R), infx∈R3 V (x) > 0. (V2) For any b > 0 such that the set {x ∈ R3 : V (x) < b} is nonempty and has finite Lebesgue measure. Except for (V1)–(V2), the following, (V3), is needed. (V3) Ω = int V –1(0) is nonempty and has smooth boundary and Ω = V –1(0). The potential V (x) with assumptions (V1)–(V3) are usually referred as the steep well potential. It was firstly proposed by Bartsch and Wang [2] to study a nonlinear Schrödinger equation. (V1)–(V2) are used to guarantee the compactness of the space
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