A Minmax Principle, Index of the Critical Point, and Existence of Sign-changing Solutions to Elliptic Boundary Value Problems
Date
1998-01-30
Authors
Castro, Alfonso
Cossio, Jorge
Neuberger, John W.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
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.
Description
Keywords
Dirichlet problem, Sign-changing solution
Citation
Castro, A., Cossio, J., & Neuberger, J. M. (1998). A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. <i>Electronic Journal of Differential Equations, 1998</i>(02), pp. 1-18.
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Attribution 4.0 International
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This work is licensed under a Creative Commons Attribution 4.0 International License.