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

We consider a quasilinear elliptic problem of the form -Delta pu = lambda f(u) in Omega, u = 0 on partial derivative Omega, where lambda > 0 is a parameter, 1 < p < 2 and Omega is a strictly convex bounded domain in R-N, N > p, with C-2 boundary partial derivative Omega. The nonlinearity f: [0, infinity) -> 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-infinity a priori bound, we establish the existence of a positive solution for lambda small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at lambda = 0. We also extend our study to systems.