Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains

Kondratiev, Vladimir; Liskevich, Vitali; Sobol, Zeev

doi:10.1090/S0002-9947-08-04453-X

Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains
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by Vladimir Kondratiev, Vitali Liskevich and Zeev Sobol PDF

Trans. Amer. Math. Soc. 361 (2009), 697-713 Request permission

Abstract:

We study the problem of the existence and non-existence of positive super-solutions to a semi-linear second-order non-divergence type elliptic equation $\sum _{i,j=1}^N a_{ij}(x)\frac {\partial ^2 u}{\partial x_i \partial x_j}+u^p=0$, $-\infty <p<\infty$, with measurable coefficients in exterior domains of $\mathbb {R}^N$. We prove that in a “generic” situation there is one critical value of $p$ that separates the existence region from non-existence. We reveal the quantity responsible for the qualitative picture and for the numerical value of the critical exponent which becomes available under a mild stabilization condition at infinity.

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