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Stable weak solutions of weighted nonlinear elliptic equations

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  • 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 $R^N$.
    Mathematics Subject Classification: Primary: 35J91; Secondary: 35B35, 35B53.

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