Nhảy đến nội dung

Stable solutions to weighted quasilinear problems of lane-emden type

Authors: 

Phuong Le, Vu Ho*

Source title: 
Electronic Journal of Differential Equations, 2018(71): 1-11, 2018 (ISI)
Academic year of acceptance: 
2018-2019
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.