Ground states for the double weighted critical Kirchhoff equation on the unit ball in $\mathbb{R}^3$

Abstract

This paper deals with the existence of ground states for degenerative ($a=0$) and non-degenerative ($a>0$) double weighted critical Kirchhoff equation \begin{eqnarray*} \left\{ \begin{array}{ll} \displaystyle-\left(a+b\int_B |\nabla u|^2dx\right)\Delta u=|x|^{\alpha_1} |u|^{4+2\alpha_1}u+\mu|x|^{\alpha_2} |u|^{4+2\alpha_2}u+\lambda h(|x|) f(u) &{\rm in}\ B,\\ u=0 &{\rm on}\ \partial B, \end{array} \right. \end{eqnarray*} where $B$ is a unit open ball in $\mathbb{R}^3$ with center $0$, $a\geq0, b>0, \mu\in \mathbb{R}, \lambda>0, \alpha_1>\alpha_2>-2$, $4+2\alpha_i=2^*(\alpha_i)-2\ (i=1,2)$ with $2^*(\alpha_i)=\frac{2(N+\alpha_i)}{N-2} $ $(N=3)$ being Hardy-Sobolev ($-2<\alpha_i<0$), Sobolev ($\alpha_i=0$) or H\'{e}non-Sobolev ($\alpha_i>0$) critical exponent of the embedding $H_{0,r}^1(B)\hookrightarrow L^p(B;|x|^{\alpha_i})$. Noting that the sign of $\mu$ gives rise to a great effect on the existence of solutions. The methods rely on Nehari manifold and the mountain pass theorem.