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Spike-layered solutions for an elliptic system with Neumann boundary conditions

Authors: Miguel Ramos and Jianfu Yang
Journal: Trans. Amer. Math. Soc. 357 (2005), 3265-3284
MSC (2000): Primary 35J50, 35J55, 58E05
Published electronically: November 4, 2004
MathSciNet review: 2135746
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Abstract: We prove the existence of nonconstant positive solutions for a system of the form $-\varepsilon^2\Delta u + u = g(v)$, $-\varepsilon^2\Delta v + v = f(u)$ in $\Omega$, with Neumann boundary conditions on $\partial \Omega$, where $\Omega$ is a smooth bounded domain and $f$, $g$are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of $\varepsilon$, the corresponding solutions $u_{\varepsilon}$ and $v_{\varepsilon}$ admit a unique maximum point which is located at the boundary of $\Omega$.

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Additional Information

Miguel Ramos
Affiliation: CMAF and Faculty of Sciences, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal

Jianfu Yang
Affiliation: Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P.O. Box 71010, Wuhan, Hubei 430071 People’s Republic of China

Keywords: Superlinear elliptic systems, spike-layered solutions, positive solutions, minimax methods
Received by editor(s): December 24, 2002
Received by editor(s) in revised form: December 21, 2003
Published electronically: November 4, 2004
Additional Notes: The first author was partially supported by FCT
The second author was supported by NNSF of China
Article copyright: © Copyright 2004 American Mathematical Society

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