### Critical growth elliptic problem in $\mathbb R^2$ with singular discontinuous nonlinearities

#### Abstract

Let $\varOmega$ be a bounded domain in $\mathbb R^{2}$ with smooth boundary, $a > 0, \lambda>0$ and $0 < \delta < 3$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} \begin{array}{rl} -\Delta u & = \lambda ( {\chi_{\{u < a\}}}{u^{-{\delta}}} + h(u) e^{u^2})~~\text{in} ~~\Omega, \\ u & > 0 ~\text{ in }~ \Omega,\\ u & = 0 ~\text{ on }~ \partial \Omega, \end{array} \end{eqnarray*} where $\chi$ is the characteristic function and $h(u)$ is a smooth nonlinearity that is a "perturbation" of $e^{u^2}$ as $u \to \infty$ (for precise definitions, see hypotheses (H1)-(H5) in Section 1). With these assumptions we study the existence of multiple positive solutions to the above problem.

#### Article information

Source
Adv. Differential Equations Volume 19, Number 5/6 (2014), 409-440.

Dates
First available in Project Euclid: 3 April 2014

Dhanya, R.; Prashanth, S.; Sreenadh, K.; Tiwari, Sweta. Critical growth elliptic problem in $\mathbb R^2$ with singular discontinuous nonlinearities. Adv. Differential Equations 19 (2014), no. 5/6, 409--440. https://projecteuclid.org/euclid.ade/1396558057.