| Summary: | We study positive solutions to multiparameter boundary-value problems of the form egin{gather*} - Delta u =lambda g(u)+mu f(u)quad ext{in } Omega \ u =0 quad ext{on } partial Omega , end{gather*} where $lambda >0$, $mu >0$, $Omega subseteq R^{n}$; $ngeq 2$ 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.
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