Multi-bump Solutions for a Strongly Indefinite Semilinear Schr\"odinger Equation Without Symmetry or convexity Assumptions

TL;DR

This paper proves the existence of infinitely many multi-bump solutions for a strongly indefinite semilinear Schrödinger equation with periodic coefficients, using novel variational and Morse theory techniques without symmetry assumptions.

Contribution

Introduces a new variational reduction method and generalized Morse theory to establish multiple solutions without symmetry or convexity constraints.

Findings

Existence of infinitely many solutions.

Existence of multi-bump solutions for any number of bumps.

Solutions are geometrically distinct and under certain conditions are isolated.

Abstract

In this paper, we study the following semilinear Schr\"odinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term $f(x,t)$ satisfies some superlinear growth conditions and need not be odd or increasing strictly in $t$. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are isolated, then we can show that this equation has infinitely many $m-$bump solutions for any positive integer $m\geq 2.$