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)Δu=|x|^{α_1} |u|^{4+2α_1}u+μ|x|^{α_2} |u|^{4+2α_2}u+λh(|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 $0$, $a\ge 0,b>0,\mu \in \mathbb{R},\lambda >0,{\alpha }_1>{\alpha }_2>-2$, $4+2{\alpha }_i=2^{\ast }({\alpha }_i)-2(i=1,2)$ with $2^{\ast }({\alpha }_i)=\frac{2(N+{\alpha }_i)}{N-2}$ $(N=3)$ being Hardy-Sobolev ($-2<{\alpha }_i<0$), Sobolev (${\alpha }_i=0$) or Hénon-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.