On the Fredholm alternative for the $p$-Laplacian

Abstract:

Consider \begin{equation*}\left \{\begin {split} &-(|u’|^{p-2}u’)’=\lambda |u|^{p-2}u+f(x), x\in (0, 1),\ &u(0)=\beta u’(0), \quad u’(1)=0,\end{split}\right . \end{equation*} where $p>1$ and $\beta \in \mathbb {R}\cup \{\infty \}$ and let $\lambda _{1}$ be the principal eigenvalue of the problem with $f(x)\equiv 0$. For $\lambda =\lambda _{1}$, we discuss for which values of $p$ and $\beta$ the Fredholm alternative holds.