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On concentration of solution to a Schrödinger logarithmic equation with deepening potential well
Author(s) -
Alves Claudianor O.,
Morais Filho Daniel C.,
Figueiredo Giovany M.
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5699
Subject(s) - mathematics , logarithm , limit (mathematics) , domain (mathematical analysis) , regular polygon , work (physics) , mathematical analysis , class (philosophy) , pure mathematics , geometry , quantum mechanics , physics , artificial intelligence , computer science
In this work, we prove the existence of positive solution for the following class of problems− Δ u + λ V ( x ) u = u log u 2 , x ∈ R N ,u ∈ H 1 ( R N ) ,where λ >0 and V : R N → R is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C 1 functional with a convex lower semicontinuous functional, we prove that for each large enough λ >0, there exists a positive solution for the problem, and that, as λ →+ ∞ , such solutions converge to a positive solution of the limit problem defined on the domain Ω=int( V −1 ({0})).