Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 96, pp. 1-10.

Title: Positive solutions for classes of multiparameter
       elliptic semipositone problems

Authors: Scott Caldwell (Mississippi State Univ, MS, USA)
         Alfonso Castro (Harvey Mudd College, Claremont, CA, USA)
         Ratnasingham Shivaji (Mississippi State Univ, MS, USA)
	 Sumalee Unsurangsie (Mahidol Univ., Thailand)

Abstract:
 We study positive solutions to  multiparameter
 boundary-value problems of the form
 \begin{gather*}
 - \Delta u =\lambda g(u)+\mu f(u)\quad \text{in  } \Omega \\
 u  =0 \quad  \text{on }  \partial \Omega ,
 \end{gather*}
 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.