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