# American Institute of Mathematical Sciences

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January  2013, 33(1): 123-140. doi: 10.3934/dcds.2013.33.123

## Existence and qualitative properties of solutions for nonlinear Dirichlet problems

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

Received  May 2011 Revised  October 2011 Published  September 2012

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.
Citation: Alfonso Castro, Jorge Cossio, Carlos Vélez. Existence and qualitative properties of solutions for nonlinear Dirichlet problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
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