Infinitely many positive solutions of nonlinear Schrodinger equations

TL;DR

This paper proves the existence of infinitely many positive multi-bump solutions for a class of nonlinear Schrödinger equations with specific potential conditions, extending previous results without symmetry assumptions.

Contribution

It establishes the existence of infinitely many positive solutions under new potential conditions, improving upon prior work by removing symmetry constraints.

Findings

Existence of infinitely many positive multi-bump solutions.

Conditions on potential ensure solution multiplicity.

Improves previous results by relaxing symmetry assumptions.

Abstract

The paper deals with the equation $-\Delta u+a(x) u =|u|^{p-1}u $, $u \in H^1(\mathbb{R}^N)$, with $N\ge 2$, $p>1,\ p<{N+2\over N-2}$ if $N\ge 3$, $a\in L^{N/2}_{loc}(\mathbb{R}^N)$, $\inf a>0$, $\lim_{|x| \to \infty} a(x)= a_\infty$. Assuming on the potential that $\lim_{|x| \to \infty}[a(x)-a_\infty] e^{\eta |x|}= \infty \ \ \forall \eta>0$ and $ \lim_{\rho \to \infty} \sup \left\{a(\rho \theta_1) - a(\rho \theta_2) \ :\ \theta_1, \theta_2 \in \mathbb{R}^N,\ |\theta_1|= |\theta_2|=1 \right\} e^{\tilde{\eta}\rho} = 0 \mbox{ for some } \ \tilde{\eta}>0$, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers [8,12,15,17,28].