On the numerical computation of mountain pass solutions to some perturbed semi-linear elliptic problem

In this paper we want to visualize and construct the shape of least energy solutions to a singularly perturbed problem (\({\tilde M}_\varepsilon\)) 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. We are mainly interested in using an appropriate numerical method to show the location and the shape of such type of solutions when the singular perturbation parameter goes to zero, analyzing the geometrical effect of the curved boundary of the domain.