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.