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 $0$, $a\geq0, b>0, μ\in \mathbb{R}, λ>0, α_1>α_2>-2$, $4+2α_i=2^*(α_i)-2\ (i=1,2)$ with $2^*(α_i)=\frac{2(N+α_i)}{N-2} $ $(N=3)$ being Hardy-Sobolev ($-2<α_i<0$), Sobolev ($α_i=0$) or Hénon-Sobolev ($α_i>0$) critical exponent of the embedding $H_{0,r}^1(B)\hookrightarrow L^p(B;|x|^{α_i})$. Noting that the sign of $μ$ gives rise to a great effect on the existence of solutions. The methods rely on Nehari manifold and the mountain pass theorem.

Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• Lemma 2.7
• Lemma 2.8
• Theorem 3.1
• Lemma 3.2: 2020Wang-Su1