Multi-bump solutions for a Kirchhoff problem type

TL;DR

This paper proves the existence of multi-bump solutions for a Kirchhoff problem in , using variational methods, under conditions on the potential well and nonlinear growth.

Contribution

It introduces new variational techniques to establish positive multi-bump solutions for Kirchhoff problems with potential wells.

Findings

Existence of positive multi-bump solutions in .

Solutions localized in disjoint potential well components.

Application of variational methods to nonlinear Kirchhoff equations.

Abstract

In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} \lambda a(x)+1)u^{2} dx\biggl) \biggl(- \Delta u + (\lambda a(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3}, \\ \mbox{}\\ u \in H^{1}(\mathbb{R}^{3}). \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well with $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.