Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sign

Let Omega subset of R-N be a bounded domain. We consider the nonlinear problem -Deltan = u(P) + lambdaf(x), x is an element of Omega, u = 0, x is an element of partial derivativeOmega, and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative, solutions of the following linear problem: -Deltav = f(x), x is an element of Omega, v = 0, x is an element of partial derivativeOmega. In particular, if p > (N + 2)/(N - 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in 1 of this paper).