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

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^N$, $N\ge 3$ with smooth boundary, $a>0,\lambda >0$ and $0<\delta <3$ be real numbers. Define $2^{\ast }:=\frac{2N}{N-2}$ 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} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u<a\}}u^{-\de} \right), u > 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to the above problem.

23 pages, Complex variables and elliptic equations, 2016