Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N

Abstract

We prove a Liouville-type theorem for stable solution of the singular quasilinear elliptic equations $$\displaylines{ -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad \text{in } \mathbb{R}^N, \cr -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad \text{in } \mathbb{R}^N }$$ where $2\le p-p(1+a)$ and $c_0>0$. The results hold for $1\le p-1<q=q_c(p,N,a,b)$ in the first equation, and for $2\le N<q_0(p,a,b)$ in the second equation. Here $q_0$ and $q_c$ are exponents, which are always larger than the classical critical ones and depend on the parameters a,b.

Keywords

• Singular quasilinear elliptic equation
• stable solutions
• critical exponents
• Liouville type theorems