Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular   Discontinuous Nonlinearities
  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. 
R. Dhanya, S. Prashanth, Sweta Tiwari and K. Sreenadh
math.AP