Geodesic
A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems
Electronic Journal of Differential Equations, Tome 1998 (1998).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35J20, 35J25, 35J60
Keywords: Dirichlet problem, sign-changing solution 
@article{EJDE_1998__1998__a13,
     author = {Castro, Alfonso and Cossio, Jorge and Neuberger, John M.},
     title = {A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {1998},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a13/}
}
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%J Electronic Journal of Differential Equations
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Castro, Alfonso; Cossio, Jorge; Neuberger, John M. A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. Electronic Journal of Differential Equations, Tome 1998 (1998). http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a13/