Abstract
In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: \t\t\t−Δu+V(x)u=[|x|−μ∗|u|p]|u|p−2u,x∈RN,\\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} $$\\begin{aligned} -\\Delta u+V(x)u=\\bigl[ \\vert x \\vert ^{-\\mu }\\ast \\vert u \\vert ^{p}\\bigr] \\vert u \\vert ^{p-2}u,\\quad x \\in \\mathbb{R}^{N}, \\end{aligned}$$ \\end{document} where Ngeq 3, 0<mu <N, frac{2N-mu }{N}leq p<frac{2N-mu }{N-2}, ∗ represents the convolution between two functions. We assume that the potential function V(x) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.
Highlights
Introduction and main resultConsider the following Choquard equation:– u + V (x)u = |x|–μ ∗ |u|p |u|p–2u, x ∈ RN, (1.1) whereProblem arises from the study of the existence of standing wave solutions for the following equation: iψt = – ψ + W ψ – |x|–μ ∗ |ψ|p |ψ|p–2ψ, which appears naturally in optical systems with a thermal [21] and influences the propagation of electromagnetic waves in plasmas [2] and plays an important role in the theory of Bose–Einstein condensation [9]
In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation:
Mathematical work on nonlinear Choquard equations like the above has been investigated in recent years, and the existence and multiplicity results for such type equations have been considered in many papers under some different assumptions on the potential and nonlinearity by using various variational arguments
Summary
In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: Mathematical work on nonlinear Choquard equations like the above has been investigated in recent years, and the existence and multiplicity results for such type equations have been considered in many papers under some different assumptions on the potential and nonlinearity by using various variational arguments. Up to translations, Lieb [18] obtained the existence and uniqueness of the ground state solutions with V being a positive constant.
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