Existence and qualitative properties of solutions for nonlinear Dirichlet problems

Alfonso Castro; Jorge Cossio; Carlos Vélez

doi:10.3934/dcds.2013.33.123

Article Contents

2013, Volume 33, Issue 1: 123-140. Doi: 10.3934/dcds.2013.33.123

This issue Previous Article Lyapunov inequalities for partial differential equations at radial higher eigenvalues Next Article On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian

Existence and qualitative properties of solutions for nonlinear Dirichlet problems

1.
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
2.
Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellĺn, Apartado Aéreo 3840, Medellín
3.
Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Apartado Aéreo 3840, Medellín

Received: May 2011

Revised: October 2011

Published: January 2013

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Abstract

sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

Keywords:

Mathematics Subject Classification: 35J20, 35J60, 35B38.

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