Multiple interior peak solutions for some singularly perturbed Neumann problems. (English) Zbl 1061.35502
Summary: We consider the problem , in on , where is a bounded smooth domain in is a small parameter, and is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as approaches zero, at a critical point of the mean curvature function . It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function . In this paper, we prove the existence of interior -peak solutions at the local maximum points of the following function: . We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function appears naturally in the asymptotic expansion of the energy functional.
|35J25||Second order elliptic equations, boundary value problems|
|35B25||Singular perturbations (PDE)|
|47J30||Variational methods (nonlinear operator equations)|
|47N20||Applications of operator theory to differential and integral equations|