Electron. J. Diff. Eqns., Vol. 1998(1998), No. 02, pp. 1-18.

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

Alfonso Castro, Jorge Cossio, John M. Neuberger

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.

Submitted September 17, 1997. Published January 30, 1998.
Math Subject Classification: 35J20, 35J25, 35J60.
Key Words: Dirichlet problem, sign-changing solution.

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Alfonso Castro
Department of Mathematics, University of Texas
San Antonio, TX 78249 USA
e-mail: castro@math.utsa.edu
Jorge Cossio
Departamento de Matematicas
Universidad Nacional de Colombia
Apartado Aereo 3840, Medellin, Colombia
e-mail: jcossio@perseus.unalmed.edu.co
John M. Neuberger
Department of Mathematics
Northern Arizona University
Flagstaff, AZ 86011 USA
e-mail: John.Neuberger@nau.edu

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