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Infinitely many solutions for a Schrodinger equation with sign-changing potential and nonlinear term

Long-Jiang Gu, Huan-Song Zhou

Abstract

We propose a new variational approach to finding multiple critical points for strongly indefinite problems without assuming the weak upper semicontinuity on the variational functionals. By this approach, we obtain the existence of infinitely many geometrically distinct solutions for a stationary periodic Schrödinger equation, in which the linear part is strongly indefinite and the nonlinear term is allowed to change sign in general ways.

Infinitely many solutions for a Schrodinger equation with sign-changing potential and nonlinear term

Abstract

We propose a new variational approach to finding multiple critical points for strongly indefinite problems without assuming the weak upper semicontinuity on the variational functionals. By this approach, we obtain the existence of infinitely many geometrically distinct solutions for a stationary periodic Schrödinger equation, in which the linear part is strongly indefinite and the nonlinear term is allowed to change sign in general ways.

Paper Structure

This paper contains 4 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two deformation lemmas
  4. Proof of Theorem \ref{['5.main result']}

Key Result

Theorem 1.1

If the conditions $(\mathbf{V})$ and $\mathbf{(f_1)}$-$\mathbf{(f_5)}$ are satisfied, then equation (5.equation) has infinitely many geometrically distinct solutions.

Theorems & Definitions (25)

  • Remark 1.1
  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 15 more