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Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

Linlian Xiao, Jiaqian Yuan, Jian Zhou, Yunshun Wu

Abstract

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)Δu+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\ u\in H^1(\mathbb{R}^3), \end{array} \right. \] where $a,b$ are postive constants, and the potential $V(x)$ is continuous and indefinite in sign. Under some suitable assumptions on $V(x)$ and $f$, we obtain the existence of solutions by the Symmetric Mountain Pass Theorem.

Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

Abstract

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)Δu+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\ u\in H^1(\mathbb{R}^3), \end{array} \right. \] where are postive constants, and the potential is continuous and indefinite in sign. Under some suitable assumptions on and , we obtain the existence of solutions by the Symmetric Mountain Pass Theorem.
Paper Structure (3 sections, 4 theorems, 41 equations)

This paper contains 3 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. Proof of main results

Key Result

Theorem 1.1

Under assuptions $(V_1),~(V_2)$ and $(f_1)-(f_4)$, if $f(x,u)$ is odd in $u$, then problem (e1-1) possesses infinitely many solutions.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof