On the shape of least‐energy solutions to a semilinear Neumann problem

Exact multiplicity results for a singularly perturbed Neumann problem

“…This problem was first studied in the papers [12], [14], [15], where the authors analyzed the asymptotic behavior of the least energy solution of the functional naturally associated. Among other results, they proved that for ε small enough the least energy solution to (1.1) has exactly one local maximum and concentrates at a point P which achieves the maximum of the mean curvature of the boundary of Ω.…”

“…(6.10) We want to pass to the limit in (6.8). To do this we first observe that by definition |φ n | 1 and also max{|∇φ n | ; ∂Ω εn } is uniformly bounded (to check the last claim one can use the diffeomorphism which straightens the boundary portion near a point P ∈ Ω, as in [12,14,15], then apply Schauder's estimates near the boundary, see for example [6,Chapter 6], to the corresponding elliptic equation satisfied in a half ball as well as the C 2,α uniform regularity of the domain). At this point, using (6.10) and Lebesgue's Dominated Convergence Theorem, we can proceed as in the proof of Proposition 5.4 to pass to the limit in (6.8) and conclude that…”

“…It is easy to prove that these solutions U ξε + w(ε, ξ ε ) are boundary single peak solutions, i.e., they have exactly one local maximum point in Ω ε which lies on ∂Ω ε . See [10, Lemma 4.2] or the arguments used in [14].…”

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

“…Our motivation for the study of such a problem goes back to the pioneering work of Ni and Takagi (see e.g. [13]) concerning the single equation −ε 2 ∆u + u = f (u) in Ω.…”

“…in [12], [18], [9] (where a "local" version of Ni-Wei's result is proved) and in [2], [6], [11] (where a linear term V (x)u is added to equation (1.5) while Ω = R N ). The subject was revisited by Del Pino and Felmer, for both Neumann and Dirichlet boundary conditions, in [5], where shorter and more elementay arguments were introduced, with respect to those in [13,14]. We refer the reader to the nice Introduction in [5] for further references and a developed discussion on the subject.…”