Electron. J. Differential Equations, Vol. 2018 (2018), No. 81, pp. 1-11.

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

Caisheng Chen, Hongxue Song, Hongwei Yang

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 $2\le p<N, -\infty<a<(N-p)/p$ and the function f(x) is continuous and nonnegative in $\mathbb{R}^N\setminus\{0\}$ such that $f(x)\ge c_0|x|^{b}$ as $|x|\ge R_0$, with $b>-p(1+a)$ 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.

Submitted June 26, 2017. Published March 22, 2018.
Math Subject Classifications: 35J60, 35B53, 35B33, 35B45.
Key Words: Singular quasilinear elliptic equation; stable solutions; critical exponents; Liouville type theorems.

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Caisheng Chen
College of Science, Hohai University
Nanjing 210098, China
email: cshengchen@hhu.edu.cn
Hongxue Song
College of Science, Hohai University
Nanjing 210098, China
email: songhx@njupt.edu.cn
Hongwei Yang
College of Mathematics and System Science
Shandong University of Science and Technology
Qingdao 266590, China
email: hwyang1979@163.com

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