Existence of multiple solutions and estimates of extremal values for a Kirchhoff type problem with fast increasing weight and critical nonlinearity

Abstract

In this article, we study the Kirchhoff type problem $$ -\Big(a+\epsilon\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\Big)\hbox{div} (K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, $$ where $x\in \mathbb{R}^3$, $10$ is small enough, and the parameters $a, \lambda >0$. Under some assumptions on $f(x)$, we establish the existence of two nonnegative nontrivial solutions and obtain uniform lower estimates for extremal values of the problem via variational methods.

Keywords

• Variational methods
• Kirchhoff type equation
• critical nonlinearity
• multiple solutions
• extremal values