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

TL;DR

This paper establishes the existence of ground states for a class of weighted critical Kirchhoff equations on the unit ball in three dimensions, analyzing both degenerative and non-degenerative cases with variational methods.

Contribution

It provides new existence results for ground states of weighted critical Kirchhoff equations, considering the influence of weights and the sign of parameters using Nehari manifold and mountain pass techniques.

Findings

Existence of ground states depends on the sign of parameter μ.

Results cover degenerative (a=0) and non-degenerative (a>0) cases.

Application of variational methods to weighted critical problems.

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.