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

Abstract

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(\ast )$: $-\nabla \cdot a\cdot \nabla u=u^p$ in an unbounded cone--like domain $G\subset {\mathbf{R}}^{\mathbf{N}}$ $(N\ge 3)$. We prove that the critical exponent $p^{\ast }(a,G)=\inf \{p>1:(\ast )has\; a\; positive\; supersolution\; inG\}$ 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.

20 pages

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