Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

Abstract

We study a perturbed semilinear problem with Neumann boundary conditionwhere Ω is a bounded smooth domain of R N , N ≥ 2, ε > 0, 1 < p < N +2

Boundary concentration phenomena for a singularly perturbed elliptic problem

“…Mountain pass solutions, for example, possess a maximum point M ε at ∂ such that H (M ε ) → max M∈∂ H (M), where H is the mean curvature of ∂ . This result as well as the uniqueness of M ε was established in [32,33] and motivated the search of more general solutions of (P ε ) concentrating at critical points of H ; see [12,17,18,19,26,31,34,41].…”

Triple Junction Solutions for a Singularly Perturbed Neumann Problem

“…It has been proved that higher energy solutions exist, which concentrates at one or several points of the boundary, or at one or more points in the interior, or a combination of the two effects. See [5], [4], [11]- [8], [20]- [17], [23]- [24], [39], [45]- [46]- [26] and the references therein. In particular, Lin, Ni and Wei [26] showed that there are at least C N ( | log |) N number of interior spikes.…”

A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture

“…As in the subcritical case the least energy solution blows up, as μ goes to infinity, at a unique point which maximizes the mean curvature of the boundary [3], [42]. Higher energy solutions have also been exhibited, blowing up at one [2], [55], [48], [26] or several separated boundary points [41], [37], [56], The above conjecture was studied by Adimurthi-Yadava [4], [5] and Budd-KnappPeletier [11] in the case Ω = B R (0) and u radial. Namely, they considered the following problem: Theorem A ( [4], [5], [6], [11]).…”