Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Details Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

2003-07-10

arxiv.org/abs/math/0307144v2

Mathematics

Analysis of PDEs

20 pages

Scientific paper

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.

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