Multi-bump solutions for Choquard equation with deepening potential well

TL;DR

This paper proves the existence of multiple localized solutions to a nonlinear Choquard equation with a deepening potential well, showing that the number of solutions increases exponentially with the number of potential well components.

Contribution

It establishes the existence of at least 2^k - 1 multi-bump solutions for large enough potential well depth in a Choquard equation with multiple disjoint potential wells.

Findings

Existence of multi-bump solutions for large λ

Number of solutions grows exponentially with potential well components

Solutions are localized in different potential well regions

Abstract

We study the existence of multi-bump solutions to Choquard equation $$ \begin{array}{ll} -\Delta u + (\lambda a(x)+1)u=\displaystyle\big(\frac{1}{|x|^{\mu}}\ast |u|^p\big)|u|^{p-2}u \mbox{ in } \,\,\, \R^3, \end{array} $$ where $\mu \in (0,3), p\in(2, 6-\mu)$, $\lambda$ is a positive parameter and the nonnegative function $a(x)$ has a potential well $ \Omega:=int (a^{-1}(0))$ consisting of $k$ disjoint bounded components $ \Omega:=\cup_{j=1}^{k}\Omega_j$. We prove that if the parameter $\lambda$ is large enough then the equation has at least $2^{k}-1$ multi-bump solutions.