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November  2010, 9(6): 1731-1752. doi: 10.3934/cpaa.2010.9.1731

On the shape of the least-energy solutions to some singularly perturbed mixed problems

Luigi Montoro 1,

1. 

Dipartimento di Matematica, UNICAL, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende, Cosenza, Italy

Received  December 2009 Revised  March 2010 Published  August 2010

In this paper we want to \emph{characterize} and \emph{visualize} the shape of some solutions to a singularly perturbed problem \eqref{eq:pe} with mixed Dirichlet and Neumann boundary conditions. Such type of problem arises in several situations as reaction-diffusion systems, nonlinear heat conduction and also as limit of reaction-diffusion systems coming from chemotaxis. In particular we are interested in showing the location and the shape of {\it least energy solutions} when the singular perturbation parameter goes to zero, analyzing the geometrical effect of the \emph{curved boundary} of the domain.
Keywords: mountain pass method, numerical solution., Singularly perturbed elliptic problems, mixed problems.
Mathematics Subject Classification: 35B25, 35J20, 35J60, 65N3.
Citation: Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731
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