On the Fredholm alternative for the 𝑝-Laplacian

Abstract

Consider { a m p ; − ( | u ′ | p − 2 u ′ ) ′ = λ | u | p − 2 u + f ( x ) , x ∈ ( 0 , 1 ) ,   a m p ; u ( 0 ) = β u ′ ( 0 ) , u ′ ( 1 ) = 0 , \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 p>1 and β ∈ R ∪ { ∞ } \beta \in \mathbb {R}\cup \{\infty \} and let λ 1 \lambda _{1} be the principal eigenvalue of the problem with f ( x ) ≡ 0 f(x)\equiv 0 . For λ = λ 1 \lambda =\lambda _{1} , we discuss for which values of p p and β \beta the Fredholm alternative holds.