Multi-peak solutions for some singular perturbation problems

Abstract

We consider the problem \[ \left \{ \begin{array}{rcl} \varepsilon^2 \Delta u - u + f(u) = 0 & \mbox{ in }& \Omega u > 0 \mbox{ in} \Omega, u = 0 & \mbox{ on }& \partial\Omega, \end{array} \right.\nonumber \] where $\Omega$ is a smooth domain in $R^N$ , not necessarily bounded, $\varepsilon >0$ is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as $\varepsilon$ approaches zero, at a maximum of the function $d(x)=d(\cdot,\partial\Omega)$ , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain $\Omega$ presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity.