Advances in Differential Equations

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

R. Dhanya, S. Prashanth, K. Sreenadh, and Sweta Tiwari

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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

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

First available in Project Euclid: 3 April 2014

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Mathematical Reviews number (MathSciNet)

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 35A01: Existence problems: global existence, local existence, non-existence 35J25: Boundary value problems for second-order elliptic equations 35J75: Singular elliptic equations


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.

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