Multiple positive solutions of singular and critical elliptic problem in $${\mathbb{R}^2}$$ with discontinuous nonlinearities

Abstract

Let Ω be a bounded domain in \({\mathbb{R}^2}\) with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where \({0\leq q a.\end{array}\right.}}\) Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.