Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular Discontinuous Nonlinearities

TL;DR

This paper investigates the existence and multiplicity of solutions for a critical elliptic problem in a bounded domain involving singular and discontinuous nonlinearities, expanding understanding of such complex differential equations.

Contribution

It introduces new results on the existence and multiplicity of solutions for a critical elliptic problem with singular and discontinuous nonlinearities.

Findings

Existence of solutions under certain parameter conditions

Multiple solutions established for specific parameter ranges

Analysis of solution behavior near singularities and discontinuities

Abstract

Let $\Omega$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, \lambda>0$ and $0<\delta<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $\chi_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -\Delta u &= \lambda \left(u^{2^*-1}+ \displaystyle \chi_{\{u<a\}}u^{-\de} \right), u > 0~~\text{in} ~~\Omega, \\ u & = 0 ~\text{on}~ \partial \Omega. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to the above problem.