Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

Abstract

Abstract. Let Ω be a bounded domain in R n , n ≥ 3, with a boundary ∂Ω ∈ C 2 . We consider the following singularly perturbed nonlinear elliptic problem on Ω:where the nonlinearity f is of subcritical growth. Under rather strong conditions on f, it has been known that for small ε > 0, there exists a mountain pass solution u ε of above problem which exhibits a spike layer near a maximum point of the distance function d from ∂Ω as ε → 0. In this paper, we construct a solution u ε of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f , which we believe to be almost optimal.

Zero mass case for a fractional Berestycki–Lions-type problem

“…It is worth noting that, a common approach to deal with fractional nonlocal problems, is to use the Caffarelli-Silvestre extension [17], which consists to realize a given nonlocal problem through a degenerate local problem in one more dimension by a Dirichlet to Neumann map. Anyway, in this work, we prefer to investigate the problem directly in H s (R N ) in order to adapt some useful methods and arguments developed in [15,32,37]. We would like to observe that our result is in clear accordance with that for the classical local counterpart.…”

Standing waves for nonlinear Schrödinger equations involving critical growth

“…Now let us say more on the background for problems like (1.1). In recent years, singularly perturbed problems have been widely studied by many researchers, and related results can been seen in [8][9][10]13…”

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

“…For N ≥ 3, it is shown in [7] and [25] that if there exists h = h(z, t) > 0 satisfying V (εΥ ε (B ε (t, z)))h 2 − 2F (h) < 0, then there exists a minimizer W = W (t, z) of J ε (t, z) over the Pohozaev manifold {u ∈ H 1 (R N ) \ {0} | Q ε (t, z)(u) = 0} which is a least energy solution of W − V (εΥ ε (B ε (t, z)))W + f (W ) = 0, W > 0 on R N .…”