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

Abstract

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.