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On Positive Multipeak Solutions of a Nonlinear Elliptic Problem
Author(s) -
Noussair Ezzat S.,
Yan Shusen
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070000898x
Subject(s) - bounded function , integer (computer science) , mathematics , combinatorics , operator (biology) , measurable function , laplace transform , physics , function (biology) , mathematical analysis , chemistry , biochemistry , repressor , evolutionary biology , biology , computer science , transcription factor , gene , programming language
In this paper we continue our investigation in [ 5 , 7 , 8 ] on multipeak solutions to the problem −ɛ 2 Δ u + u = Q ( x )| u | q −2 u , x ∈R N , u ∈ H 1 (R N ) (1.1) where Δ = ∑ N i =1 δ 2 /δ x 2 i is the Laplace operator in R N , 2 < q < ∞ for N = 1, 2, 2 < q < 2 N /( N −2) for N⩾3, and Q ( x ) is a bounded positive continuous function on R N satisfying the following conditions. (Q 1 ) Q has a strict local minimum at some point x 0 ∈R N , that is, for some δ > 0 Q ( x )> Q ( x 0 ) for all 0 < ∣ x − x 0 ∣ < δ. (Q 2 ) There are constants C , θ > 0 such that | Q ( x )− Q ( y )|⩽ C | x − y | θ for all ∣ x − x 0 ∣ ⩽ δ, ∣ y − y 0 ∣ ⩽ δ. Our aim here is to show that corresponding to each strict local minimum point x 0 of Q ( x ) in R N , and for each positive integer k , (1.1) has a positive solution with k ‐peaks concentrating near x 0 , provided ε is sufficiently small, that is, a solution with k ‐maximum points converging to x 0 , while vanishing as ε → 0 everywhere else in R N .