Semiclassical limit for the nonlinear Klein Gordon equation in bounded domains

TL;DR

This paper investigates the existence of multiple standing wave solutions to the nonlinear Klein-Gordon equation in bounded domains, focusing on the semiclassical limit as the parameter epsilon approaches zero.

Contribution

It establishes the existence of at least cat(D) standing waves for small epsilon under certain growth conditions on the potential function W.

Findings

Existence of multiple standing waves proportional to the domain's category.

Results valid for sufficiently small epsilon.

Conditions on W ensure the solutions' existence.

Abstract

We are interested to the existence of standing waves for the nonlinear Klein Gordon equation {\epsilon}^2{\box}{\psi} + W'({\psi}) = 0 in a bounded domain D. The main result of this paper is that, under suitable growth condition on W, for {\epsilon} sufficiently small, we have at least cat(D) standing wavesfor the equation ({\dag}), while cat(D) is the Ljusternik-Schnirelmann category.