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: 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, ) and by Ackermann et al. (Milan J Math, 79(1), 221-232, ) 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. [ABSTRACT FROM AUTHOR]