Electron. J. Differential Equations, Vol. 2018 (2018), No. 71, pp. 1-11.

Stable solutions to weighted quasilinear problems of Lane-Emden type

Phuong Le, Vu Ho

Abstract:
We prove that all entire stable $W^{1,p}_{\rm loc}$ solutions of weighted quasilinear problem
$$ -\hbox{div} (w(x)|\nabla u|^{p-2} \nabla u) = f(x)|u|^{q-1}u $$
must be zero. The result holds true for $p \ge 2$ and $p-1 < q < q_c(p,N,a,b)$. Here $b > a - p$ and $q_c(p,N,a,b)$ is a new critical exponent, which is infinity in low dimension and is always larger than the classic critical one, while $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are nonnegative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for large |x|. We also construct an example to show the sharpness of our result.

Submitted July 11, 2017. Published March 15, 2018.
Math Subject Classifications: 35B53, 35J92, 35B08, 35B35.
Key Words: Quasilinear problems; stable solutions; Lane-Emden nonlinearity; Liouville theorems.

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Phuong Le
Department of Mathematical Economics
Banking University of Ho Chi Minh City, Vietnam
email: phuongl@buh.edu.vn
Vu Ho
Division of Computational Mathematics and Engineering
Institute for Computational Science
Ton Duc Thang University, Ho Chi Minh City, Vietnam
email: hovu@tdt.edu.vn

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