Fredholm Alternative for the p-Laplacian in Higher Dimensions

In this paper we characterize the set of all right-hand sides h ∈ C for which the boundary value problem pu + λ1 u p−2u = h in u =0 on∂ has at least one weak solution u ∈ W 1 p 0 . Here 1 < p < 2, and λ1 > 0 is the first eigenvalue of the p-Laplacian. In particular, we prove that for hϕ1 = 0 this problem is solvable and the energy functional Eh u = 1 p ∇u p − λ1 p u p + hu is unbounded from below. 