The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem where R + N : = { x ∈ R N : x N > 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 ( P c ). We prove that the existence and multi...

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Bibliographic Details
Published in:Mathematische annalen 2022-06, Vol.383 (1-2), p.361-397
Main Authors: Fernández, Antonio J., Weth, Tobias
Format: Article
Language:English
Subjects:
Dirichlet problem
Half spaces
Mathematics
Mathematics and Statistics
Schrodinger equation
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Summary:The present paper is concerned with the half-space Dirichlet problem where R + N : = { x ∈ R N : x N > 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 ( P c ). We prove that the existence and multiplicity of bounded positive solutions to ( P c ) depend in a striking way on the value of c > 0 and also on the dimension N . We find an explicit number c p ∈ ( 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 < c p , problem ( P c ) admits infinitely many bounded positive solutions, whereas, for c > c p , there are no bounded positive solutions to ( P c ).
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-020-02129-8