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: $$\begin{aligned} -\Delta u=u^p \quad \text{ in } \Omega _t,\quad u=0 \quad \text{ on } \partial \Omega _t. \end{aligned}$$ Here $1<p<\frac{N+2}{N-2}$ when $N\ge 3,\,1<p<\infty $ when $N=2$ and $\Omega _t$ is a tubular domain which expands as $t\rightarrow \infty $ . 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 $N\ge 2$ 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.