On the Fredholm alternative for the $p$-Laplacian
On the Fredholm alternative for the $p$-Laplacian
Consider \begin{equation*}\left \{\begin {split} &-(|uâ|^{p-2}uâ)â=\lambda |u|^{p-2}u+f(x), x\in (0, 1),\ &u(0)=\beta uâ(0), \quad uâ(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 …