A counterexample to the Fredholm alternative for the p-Laplacian

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional p-Laplacian with 1 < p < infinity is considered: (*) -(\u'\(p-2)u')' - lambda\u\(p-2)u = f(x) for 0 < x < T; u(0) = u(T) = 0; where f = 1 + h with h is an element of L infinity(0, T) small enough. Solvability properties of Problem (*) with respect to the spectral parameter lambda is an element of R are investigated. We focus our attention on some fundamental differences between the cases p not equal 2 and p = 2. For p not equal 2 we give a counterexample to the classical Fredholm alternative (which is valid for the linear case p = 2).