Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains

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In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: -Delta u = u(p) in Omega(t) u=0 on partial derivative Omega(t). Here 1 < p < N+2/N-2 when N >= 3, 1 < p < infinity when N = 2 and Omega(t) and is a tubular domain which expands as t -> infinity. See (1.6) below for a precise definition of expanding tubular domain. When the section D of Omega(t) is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1-2), 23-55, 2002) and by Ackermann et al. (Milan J Math, 79(1), 221-232, 2011) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all N >= 2 without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.
Publisher
SPRINGER
Issue Date
2014-05
Language
English
Article Type
Article
Keywords

STRIP-LIKE DOMAINS; EQUATIONS; SYMMETRY; EXISTENCE; PRINCIPLE

Citation

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, v.50, no.1-2, pp.365 - 397

ISSN
0944-2669
DOI
10.1007/s00526-013-0639-z
URI
http://hdl.handle.net/10203/189930
Appears in Collection
MA-Journal Papers(저널논문)
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