Ground state solutions for the nonlinear Schrödinger–Maxwell equations

By Antonio Azzollini, Alessio Pomponio. 2007

486 citations recorded by the source corpus.

Abstract In this paper we study the nonlinear Schrodinger–Maxwell equations { − Δ u + V ( x ) u + ϕ u = | u | p − 1 u in R 3 , − Δ ϕ = u 2 in R 3 . If V is a positive constant, we prove the existence of a ground state solution ( u , ϕ ) for 2 p 5 . The non-constant potential case is treated for 3 p 5 , and V possibly unbounded below. Existence and nonexistence results are proved also when the nonlinearity exhibits a critical growth.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25c526f0451ff2134a71f253fb50e104132427a1

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In Applied Mathematics

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Citations in the corpus

Sign-changing solutions for the nonlinear Schrödinger–Poisson system in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2014, unknown): “7 of [3] we know that there exists a positive constant C such that 〈G ′(u), u〉 ≤ −C < 0, that is, 〈G ′(u), u〉 = 0, for all u ∈ Nλ.”
On the Schrödinger-Maxwell equations under the effect of a general nonlinear term (2009, background): “The greatest part of the literature focuses on the study of the previous system for the very special nonlinearity g(u) = −u + |u|u, and existence, nonexistence and multiplicity results are provided in many papers for this particular problem (see [1, 2, 10, 12, 13, 14, 19, 20, 21, 24, 28]).”
Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials (2017, methodology): “Different from [30], Li and Ye [23] worked in H1(R3) as in [3] instead of H1 r (R 3), and this means that the method used in [24] cannot be applied.”
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials (2017, methodology): “[3] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math.”
Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains (2013, background): “[5] A. Azzollini & A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math.”
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $R^3$ (2013, methodology): “In [7], Azzollini and Pomponio used the same manifold as in [30] and the concentrationcompactness argument to prove the existence of positive ground state solutions to (1.”
Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential (2015, background): “There has been a vast literature on the study of existence and multiplicity of solutions of system (SP )λ under various hypotheses on the potential V (x) and the nonlinearity f(x, u), see [1-3, 5, 9-14, 18, 19, 21, 22, 24-28, 31, 34, 35-37] and the references therein.”
Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent (2020, methodology): “Motivated by [1, 10, 20, 26], we choose the usual Sobolev spaceH1(R3) to prove the existence of ground state solutions for the “limit” problem”