Electron. J. Diff. Eqns., Vol. 2007(2007), No. 96, pp. 1-10.

Positive solutions for classes of multiparameter elliptic semipositone problems

Scott Caldwell, Alfonso Castro, Ratnasingham Shivaji, Sumalee Unsurangsie

Abstract:
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.

Submitted November 13, 2006. Published June 29, 2007.
Math Subject Classifications: 35J20, 35J65.
Key Words: Positive solutions; multiparameters; mountain pass lemma; sub-super solutions; semipositone.

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Scott Caldwell
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA
email: pscaldwell@yahoo.com
Alfonso Castro
Department of Mathematics
Harvey Mudd College
Claremont, CA 91711, USA
email: castro@math.hmc.edu
R. Shivaji
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA
e-mail: shivaji@math.msstate.edu
  Sumalee Unsurangsie
Mahidol University, Thailand

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