Multi-bump solutions for Choquard equation with deepening potential well

Abstract

We study the existence of multi-bump solutions to Choquard equation $$ \begin{array}{ll} -Δu + (λa(x)+1)u=\displaystyle\big(\frac{1}{|x|^μ}\ast |u|^p\big)|u|^{p-2}u \mbox{ in } \,\,\, \R^3, \end{array} $$ 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.

26pages

References in corpus (2)

• Nodal solutions for the Choquard equation
• Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Cited by in corpus (1)

• Existence of solutions for a class of nonlinear Choquard equations with critical growth