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

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