Boundary Singularities on a Wedge-like Domain of a Semilinear Elliptic Equation

Abstract

Let $n\ge 2$ and $\Omega \subset {\mathbb{R}}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u+u^p=0in\Omega ,$$ which vanish in a suitable trace sense on $\partial \Omega ,$ but which are singular at prescribed "edge" of $\Omega$ if $p$ is equal or slightly above a certain exponent $p_0>1$ which depends on $\Omega .$ Moreover, in the case which $\Omega$ is unbounded, the solutions have fast decay at infinity.