A counterexample to the Fredholm alternative for the $p$-Laplacian

Abstract:

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional $p$-Laplacian with $1 < p < \infty$ is considered: \begin{equation*} - (|u’|^{p-2} u’)’ - \lambda |u|^{p-2} u = f(x) \;\mbox { for } 0 < x < T ; \quad u(0) = u(T) = 0 , \tag {*}\end{equation*} 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 $p=2$).