# American Institute of Mathematical Sciences

January  2014, 13(1): 1-73. doi: 10.3934/cpaa.2014.13.1

## High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems

 1 Department of Applied Mathematics, Complutense University of Madrid, Madrid, 28040, Spain 2 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain 3 Dipartimento di Matematica e Informatica, Università, Via Delle Scienze 206, I-33100 Udine

Received  October 2012 Revised  May 2013 Published  July 2013

This paper analyzes the existence and structure of the positive solutions of a very simple superlinear indefinite semilinear elliptic prototype model under non-homogeneous boundary conditions, measured by $M\leq \infty$. Rather strikingly, there are ranges of values of the parameters involved in its setting for which the model admits an arbitrarily large number of positive solutions, as a result of their fast oscillatory behavior, for sufficiently large $M$. Further, using the amplitude of the superlinear term as the main bifurcation parameter, we can ascertain the global bifurcation diagram of the positive solutions. This seems to be the first work where these multiplicity results have been documented.
Citation: Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1
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