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2005 Multiplicity of solutions to a nonhomogeneous elliptic equation in $\Bbb R^2$
S. Prashanth, K. Sreenadh
Differential Integral Equations 18(6): 681-698 (2005).

Abstract

In this article, we consider the following problem $$(P)\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta u & = & \mu u|u|^{p}e^{u^{2}} +\lambda h(x)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; \Omega, \\ u\;\;=\;\; 0 \;\;\text{on}\;\; \partial \Omega,\hspace{1.355cm} \end{array}\right.\hspace{2cm} $$ where $0\le p<\infty, \mu,\lambda >0, \Omega\subset \mathbb R^{2}$ is a bounded domain and $h \geq 0$ in $\Omega$ with $\| h \|_{L^{2}(\Omega)}=1.$ We show that there exist real numbers $0<\lambda_{*} \le \lambda^{*}$ such that the above problem admits at least two solutions for all $\lambda\in (0,\lambda_{*})$ and no solution for $\lambda > \lambda^{*}$.

Citation

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S. Prashanth. K. Sreenadh. "Multiplicity of solutions to a nonhomogeneous elliptic equation in $\Bbb R^2$." Differential Integral Equations 18 (6) 681 - 698, 2005.

Information

Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35154
MathSciNet: MR2136705

Subjects:
Primary: 35J65
Secondary: 35J20 , 47J30 , 58E05

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.18 • No. 6 • 2005
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