Anisotropic semipositone quasilinear problems

We present the existence of positive solutions for a degenerate (or singular) quasilinear equation in a ball or in the entire space RN. As differential operator we consider a weighted p-Laplacian with a continuous coefficient satisfying a mild integrability condition towards the boundary of the ball or at infinity in RN. Our main result applies when the nonlinearities f(u) have (p−1)-sublinear growth at infinity and singular semipositone structure at the origin.