On the location of concentration points for singularly perturbed elliptic equations

Abstract

By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$, we get some necessary conditions for locating the peak-points of a class of singularly perturbed quasilinear elliptic problems in divergence form. More precisely, we show that the points where the concentration occurs, in general, must belong to what we call the set of weak-concentration points. Finally, in the semilinear case, we provide a new necessary condition which involves the Clarke subdifferential of the ground-state function.