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

## TL;DR

This paper constructs positive weak solutions to a semilinear elliptic PDE in wedge-like domains that are singular at the boundary edge, with solutions exhibiting specific decay properties depending on the domain's boundedness.

## Contribution

It introduces a method to construct solutions with boundary singularities in wedge-like domains for certain nonlinear exponents, extending understanding of boundary behavior.

## Key findings

- Solutions are singular at the boundary edge for p ≥ p_0.
- Solutions decay rapidly at infinity in unbounded domains.
- The construction depends on the geometry of the wedge-like domain.

## Abstract

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.