Existence Results for Classes of Sublinear Semipositone Problems

Abstract

We consider the semipositone problem

$${\matrix {-\Delta u (x)= \lambda f (u(x))\ \ \; \ \ \ \ \ x \in \Omega \cr \qquad \qquad \qquad u(x)=0 \ \ \ \;\ \ \ \ x \in \partial \Omega \cr}}$$

where λ > 0 is a constant, Ω is a bounded region in Rⁿ with a smooth boundary, and f is a smooth function such that f ′(u) is bounded below, f (0) < 0 and ${\rm lim}_{u \rightarrow}+\infty {f(u)\over u}=0.$ We prove under some additional conditions the existence of a positive solution (1) for λ ∈ I where I is an interval close to the smallest eigenvalue of —Δ with Dirichlet boundary condition and (2) for λ large. We also prove that our solution u for λ large is such that∥u∥ ≔ sup_x∈Ω ¦u(x)¦ → ∞ as A → ∞. Our methods are based on sub and super solutions. In particular, we use an anti maximum principle to obtain a subsolution for our existence result for λ ∈ I.