Communications on Pure and Applied Analysis (CPAA)

Stable weak solutions of weighted nonlinear elliptic equations

Pages: 293 - 305, Volume 13, Issue 1, January 2014      doi:10.3934/cpaa.2014.13.293

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Xia Huang - Department of Mathematics and Center for Partial Differential Equations, East China Normal University, Shanghai, 200241, China (email)

Abstract: This paper deals with the weighted nonlinear elliptic equation \begin{eqnarray} -\mathrm{div}(|x|^\alpha \nabla u )=|x|^\gamma e^u \ in\ \Omega ,\\ u = 0 \ on \ \partial \Omega, \end{eqnarray} where $\alpha, \gamma \in R$ satisfy $N + \alpha > 2$ and $\gamma - \alpha > -2$, and the domain $\Omega \subset R^N (N \geq 2)$ is bounded or not. Moreover, when $\alpha\neq 0$, we prove that, for $N + \alpha > 2$, $\gamma - \alpha \leq -2$, the above equation admits no weak solution. We also study Liouville type results for the equation in $R^N$.

Keywords:  Stability, weak solution, weighted sobolev space, Liouville theorems, exponential nonlinearity.
Mathematics Subject Classification:  Primary: 35J91; Secondary: 35B35, 35B53.

Received: December 2012;      Revised: April 2013;      Available Online: July 2013.