On the location of concentration points for singularly perturbed elliptic equations. (English) Zbl 1387.35258

Secchi, Simone; Squassina, Marco

Adv. Differ. Equ. 9, No. 1-2, 221-239 (2004).

Summary: 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.

MSC:

35J60	Nonlinear elliptic equations
35B25	Singular perturbations in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces