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

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ 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.