TY  - JOUR
A1  - Fernández, Antonio J.
A1  - Weth, Tobias
T1  - The nonlinear Schrödinger equation in the half-space
T2  - Mathematische Annalen
N2  - The present paper is concerned with the half-space Dirichlet problem [...] where ℝ𝑁+:={𝑥∈ℝ𝑁:𝑥𝑁>0} for some 𝑁≥1 and 𝑝>1, 𝑐>0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (𝑃𝑐). We prove that the existence and multiplicity of bounded positive solutions to (𝑃𝑐) depend in a striking way on the value of 𝑐>0 and also on the dimension N. We find an explicit number 𝑐𝑝∈(1,𝑒√), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions 𝑁≥2, we prove that, for 0<𝑐<𝑐𝑝, problem (𝑃𝑐) admits infinitely many bounded positive solutions, whereas, for 𝑐>𝑐𝑝, there are no bounded positive solutions to (𝑃𝑐).
Y1  - 2021
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/74742
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-747421
SN  - 1432-1807
N1  - Open Access funding enabled and organized by Projekt DEAL.
VL  - 383
IS  - 1–2
SP  - 361
EP  - 397
PB  - Springer
CY  - Berlin ; Heidelberg
ER  -