Concentration of positive ground state solutions for critical Kirchhoff equation with competing potentials

TL;DR

This paper studies the existence and concentration behavior of positive ground state solutions for a singularly perturbed Kirchhoff equation with competing potentials, using variational methods to analyze how solutions localize as the perturbation parameter approaches zero.

Contribution

It establishes the existence of positive ground state solutions for the Kirchhoff equation with critical growth and identifies their concentration points based on potential functions.

Findings

Existence of positive ground state solutions proven.

Concentration points of solutions identified as the perturbation parameter tends to zero.

Solutions localize in regions dictated by the potentials.

Abstract

In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{equation*} -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x)u=P(x)|u|^{p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{R}^3, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b > 0$ are constants, $p\in(4,6)$ and $V, P, Q$ are potential functions satisfying some competing conditions. We prove the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials $V,P$ and $Q$ as the concentration position of these ground state solutions as $\varepsilon\to0$.