Positive solutions for classes of multiparameter elliptic semipositone problems

Caldwell, Scott ; Castro, Alfonso ; Shivaji, Ratnasingham ; Unsurangsie, Sumalee

Electronic journal of differential equations, Tome 2007 (2007)

Résumé

We study positive solutions to multiparameter boundary-value problems of the form

$\displaylines{ - \Delta u =\lambda g(u)+\mu f(u)\quad \hbox{in } \Omega \cr u =0 \quad \hbox{on } \partial \Omega , }$

where $\lambda >0, \mu >0, \Omega \subseteq R^{n}; n\geq 2$ is a smooth bounded domain with $\partial \Omega $ in class $C^{2}$ and $\Delta $ is the Laplacian operator. In particular, we assume $g(0)>0$ and superlinear while $f(0)0$, sublinear, and eventually strictly positive. For fixed $\mu$, we establish existence and multiplicity for $\lambda $ small, and nonexistence for $\lambda $ large. Our proofs are based on variational methods, the Mountain Pass Lemma, and sub-super solutions.

Détail

Classification : 35J20, 35J65
Keywords: positive solutions, multiparameters, mountain pass lemma, sub-super solutions, semipositone