Electron. J. Differential Equations, Vol. 2016 (2016), No. 242, pp. 1-11.

Stability for noncoercive elliptic equations

Shuibo Huang, Qiaoyu Tian, Jie Wang, Jia Mu

Abstract:
In this article, we consider the stability for elliptic problems that have degenerate coercivity in their principal part,
$$\displaylines{ -\text{div}\Big(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}}\Big) +|u|^{q-1}u=f,\quad x\in\Omega, \cr u(x)=0,\quad x\in \partial\Omega, }$$
where $\theta>0$, $\Omega\subseteq \mathbb{R}^N$ is a bounded domain. Let K be a compact subset in $\Omega$ with zero r-capacity ( $p<r\leq N$). We prove that if $f_n$ is a sequence of functions which converges strongly to f in $ L^1_{\rm loc}(\Omega\backslash K)$ and $q>r(p-1)[1+\theta(p-1)]/(r-p)$, and $u_n$ is the sequence of solutions of the corresponding problems with datum $f_n$. Then $u_n$ converges to the solution u.

Submitted April 29, 2016. Published September 5, 2016.
Math Subject Classifications: 37K45，35J60.
Key Words: Removable singularity; capacity; noncoercive elliptic equation.

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Shuibo Huang
School of Mathematics and Computer
Northwest University for Nationalities
Lanzhou, Gansu 730000, China
email: huangshuibo2008@163.com
Qiaoyu Tian
School of Mathematics and Computer
Northwest University for Nationalities
Lanzhou, Gansu 730000, China
email: tianqiaoyu2004@163.com
Jie Wang
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu 730050, China
email: jiema138@163.com
Jia Mu
School of Mathematics and Computer
Northwest University for Nationalities
Lanzhou, Gansu 730000, China
email: mujia88@163.com

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