Concentration of solutions of a semilinear PDE with slow spatial dependence

Abstract

The problem $-\epsilon^2\nabla\cdot(P( x)\nabla u)+F(V( x),u)=0$ is studied in the whole of ${\mathbb R}^n$, where $V(x)$ is a multidimensional potential and $P(x)$ a matrix function. Under general conditions solutions are constructed for small positive $\epsilon$ having simple concentration properties. The asymptotic form of the solution is studied as $\epsilon\to 0$ as well as its positivity.