Abstract
This paper deals with the following Kirchhoff–Schrödinger–Poisson system: {−(a+b∫R3|∇u|2dx)Δu+V(x)u+ϕu=K(x)f(u)in R3,−Δϕ=u2in 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}$$ \\textstyle\\begin{cases} -(a+b\\int _{\\mathbb{R}^{3}} \\vert \\nabla u \\vert ^{2}\\,dx)\\Delta u+V(x)u+\\phi u=K(x)f(u)&\\text{in } \\mathbb{R}^{3}, \\\\ -\\Delta \\phi =u^{2}&\\text{in } \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where a, b are positive constants, K(x), V(x) are positive continuous functions vanishing at infinity, and f(u) is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions.
Highlights
1 Introduction In this paper, we discuss the existence of a least energy nodal solution of the following Kirchhoff–Schrödinger–Poisson system:
Where a, b are positive constants, K(x), V (x) are positive continuous functions vanishing at infinity, and f (u) is a continuous function
Inspired by the works mentioned, especially by [10, 13, 15, 35, 45], in this paper, we find the nodal solutions to system (1.1) under some weaker assumptions on f
Summary
1 Introduction In this paper, we discuss the existence of a least energy nodal solution of the following Kirchhoff–Schrödinger–Poisson system: The solution of system (1.1) is the critical point of the functional J. Lemma 2.4 Suppose that (V , K) ∈ K and (f1)–(f4) hold.
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