Multiplicity and concentration behavior of solutions to the critical Kirchhoff type problem

TL;DR

This paper investigates the existence, multiplicity, and concentration behavior of positive solutions to a critical Kirchhoff type problem in 3-dimensional space, revealing how solutions cluster around potential minima as a parameter approaches zero.

Contribution

It establishes the relationship between the number of solutions and the potential profile, and proves concentration of solutions around local minima as 3 approaches zero.

Findings

Solutions concentrate around local minima of V as 3 a small

Exponential decay of solutions at infinity

Number of solutions linked to potential profile

Abstract

In this paper, we study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)\Delta u + V(x) u = f(u)+u^5\ \ {\rm in } \ \ \R^3, \end{equation*} where $\varepsilon$ is a small positive parameter, $a$, $b$ are positive constants, $V \in C(\mathbb{R}^3)$ is a positive potential, $f \in C^1(\R^+, \R)$ is a subcritical nonlinear term, $u^5$ is a pure critical nonlinearity. When $\varepsilon>0$ small, we establish the relationship between the number of positive solutions and the profile of the potential $V$. The exponential decay at infinity of the solution is also obtained. In particular, we show that each solution concentrates around a local strict minima of $V$ as $\varepsilon \rightarrow 0$.