Geodesic
Positive solutions for classes of multiparameter elliptic semipositone problems
Electronic Journal of Differential Equations, Tome 2007 (2007).

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

Summary: 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.
Classification : 35J20, 35J65
Keywords: positive solutions, multiparameters, mountain pass lemma, sub-super solutions, semipositone 
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     author = {Caldwell, Scott and Castro, Alfonso and Shivaji, Ratnasingham and Unsurangsie, Sumalee},
     title = {Positive solutions for classes of multiparameter elliptic semipositone problems},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2007},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a24/}
}
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Caldwell, Scott; Castro, Alfonso; Shivaji, Ratnasingham; Unsurangsie, Sumalee. Positive solutions for classes of multiparameter elliptic semipositone problems. Electronic Journal of Differential Equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a24/