Multi-bump standing waves for nonlinear Schrodinger equations with a general nonlinearity: the topological effect of potential wells

TL;DR

This paper establishes the existence of multi-bump solutions for nonlinear Schrödinger equations with general nonlinearities and potential wells, using novel topological methods to handle complex potential geometries.

Contribution

It extends previous results by proving multi-bump solutions for broader classes of nonlinearities and potential wells using new topological arguments.

Findings

Multi-bump solutions exist with centers approaching potential wells as epsilon approaches zero.

The methods apply to potential wells formed by unions of submanifolds and other topological structures.

The results generalize previous work to more complex potential geometries.

Abstract

In this article, we are interested in multi-bump solutions of the singularly perturbed problem \begin{equation*} -\epsilon^2\Delta v+V(x)v=f(v) \ \ \mbox{ in }\ \ \R^N. \end{equation*} Extending previous results \cite{B, DLY,W1}, we prove the existence of multi-bump solutions for an optimal class of nonlinearities $f$ satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as $\epsilon\rightarrow 0$. Examples of potential wells include the following: the union of two compact smooth submanifolds of $\R^N$ where these two submanifolds meet at the origin and an embedded topological submanifold of $\R^N$.