Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity

Abstract

In this paper, we study the following Kirchhoff type problem \[ -\left(a+b\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\right)\hbox{div}(K(x)\nabla u)=\lambda K(x)|x|^{\beta}|u|^{q-2}u+K(x)|u|^{4}u,\quad x\in \mathbb{R}^3, \] where $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\beta=(\alpha-2)(6-q)/4$ and the parameters $a, b, \lambda >0$. When $6-\frac{4}{\alpha}0$. When $20$ small enough. In the proof we use variational methods.

Keywords

• kirchhoff type equation
• critical nonlinearity
• variational methods