Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 02, pp. 1-18.
Title: A minmax principle, index of the critical point, and existence of
sign-changing solutions to elliptic boundary value problems
Authors: Alfonso Castro (Univ. of North Texas, Denton, USA)
Jorge Cossio (Univ. Nacional de Colombia, Medellin, Colombia)
John M. Neuberger (Northern Arizona Univ., Flagstaff, USA)
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.