May/June 2014 Critical growth elliptic problem in $\mathbb R^2$ with singular discontinuous nonlinearities
R. Dhanya, S. Prashanth, K. Sreenadh, Sweta Tiwari
Adv. Differential Equations 19(5/6): 409-440 (May/June 2014). DOI: 10.57262/ade/1396558057

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.

Citation

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R. Dhanya. S. Prashanth. K. Sreenadh. Sweta Tiwari. "Critical growth elliptic problem in $\mathbb R^2$ with singular discontinuous nonlinearities." Adv. Differential Equations 19 (5/6) 409 - 440, May/June 2014. https://doi.org/10.57262/ade/1396558057

Information

Published: May/June 2014
First available in Project Euclid: 3 April 2014

zbMATH: 1294.35035
MathSciNet: MR3189090
Digital Object Identifier: 10.57262/ade/1396558057

Subjects:
Primary: 35A01 , 35J25 , 35J75 , 35J91

Rights: Copyright © 2014 Khayyam Publishing, Inc.

Vol.19 • No. 5/6 • May/June 2014
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