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

  • Jaeyoung Byeon
  • Kazunaga Tanaka
Article

Abstract

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.

Mathematics Subject Classification

35J60 35J20 58E05 

Notes

Acknowledgments

Authors are grateful to Seunghyeok Kim, Naoki Shioji and unknown referees for their helpful comments. This research of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0030749) and Mid-career Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (No. 2010-0014135). The second author is supported in part by Grant-in-Aid for Scientific Research (B)(No. 25287025) of Japan Society for the Promotion of Science.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKAISTDaejeonRepublic of Korea
  2. 2.Department of Mathematics, School of Science and EngineeringWaseda UniversityTokyoJapan

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