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

In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: (Formula presented.). Here 1 < p < N+2/N-2 when N ≥ 3, 1 < p < 8 when N = 2 and Ω_t is a tubular domain which expands as t → ∞. See (1.6) below for a precise definition of expanding tubular domain. When the section D of Ω_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.