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

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.