Multiplicity of positive solutions for a gradient system with an exponential nonlinearity
In this article, we consider the problem $$displaylines{ -Delta u = lambda u^{q} + f_1(u,v) quad hbox{in } Omegacr -Delta v = lambda v^{q} + f_{2} (u,v) quad hbox{in } Omegacr u, v > 0 quad hbox{in } Omega cr u = v = 0 quad hbox{on } partialOmega, }$$ where $Omega$ is a bounded domain in $...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2012-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/236/abstr.html |
| Summary: | In this article, we consider the problem $$displaylines{ -Delta u = lambda u^{q} + f_1(u,v) quad hbox{in } Omegacr -Delta v = lambda v^{q} + f_{2} (u,v) quad hbox{in } Omegacr u, v > 0 quad hbox{in } Omega cr u = v = 0 quad hbox{on } partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^{2}$, $0<q<1$, and $lambda>0$. We show that there exists a real number $Lambda$ such that the above problem admits at least two solutions for $lambdain(0,Lambda)$, and no solution for $lambda>Lambda$. |
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| ISSN: | 1072-6691 |