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A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems

Castro, Alfonso and Cossio, Jorge and Neuberger, John M. (1998) A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. Electronic Journal of Differential Equations, 1998 (2). pp. 1-18. ISSN 1072-6691


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Publisher’s or external URL: http://www.ams.org/mathscinet-getitem?mr=1491525


In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two sign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is sufficiently thin, and ii) the existence of arbitrarily many sign-changing non-radial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball.

Item Type: Article
Publisher’s Statement: © 1998 Southwest Texas State University and University of North Texas. This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author. This is in accordance with the BOAI definition of open access.
Keywords: Dirichlet problems; minmax principle;
Subjects: Q Science > QA Mathematics
NAU Depositing Author Academic Status: Faculty/Staff
Department/Unit: College of Engineering, Forestry, and Natural Science > Mathematics and Statistics
Date Deposited: 08 Oct 2015 17:23
URI: http://openknowledge.nau.edu/id/eprint/645

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