Existence of positive solutions for a class of p-Laplacian superlinear semipositone problems

We consider a quasilinear elliptic problem of the form where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝ N , N > p , with C 2 boundary ∂Ω . The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone ( f (0) < 0) and p -superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L ∞ a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.