A counterexample to the Fredholm alternative for the $\lowercase{p}$-Laplacian

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional $p$-Laplacian with $1 < p < \infty$ is considered:

where $f\equiv 1 + h$ with $h\in L^\infty (0,T)$ small enough. Solvability properties of Problem (*) with respect to the spectral parameter $\lambda\in \mathbb{R}$ are investigated. We focus our attention on some fundamental differences between the cases $p\neq 2$ and $p=2$. For $p\neq 2$ we give a counterexample to the classical Fredholm alternative (which is valid for the linear case $p=2$).