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A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. (English) Zbl 0901.35028

Summary: We apply the minmax principle we developed in [A. Castro, J. Cossio and J. M. Neuberger, Rocky Mt. J. Math. 27, No. 4, 1041-1053 (1997)] 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 loc. cit. with the degree-theoretic results of A. Castro and J. Cossio [SIAM J. Math. Anal. 25, 1554-1561 (1994; Zbl 0807.35039)], 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 A. Castro, J. Cossio and J. M. Neuberger [Nonlinear Anal., Theory Methods Appl. 30, No. 6, 3657-3662 (1997; Zbl 0888.35038)] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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