Existence Results for Classes of Sublinear Semipositone Problems

We consider the semipositone problem −Δu(x) = λf(u(x)) ; x є Ω u(x)=0 ; x є ∂Ω where λ \u3e 0 is a constant, Ω is a bounded region in R^n with a smooth boundary, and f is a smooth function such that f\u27(u) is bounded below, f(0) \u3c 0 and lim u→+∞f(u)/u=0. We prove under some additional conditions the existence of a positive solution (1) for λ є 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 xєΩ |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 є I