An Ambrosetti-Prodi type result for fractional spectral problems

Abstract

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-Δ_{Dir})^{s} u= f(x, u)+tφ_{1}+h &\mbox{ in } Ω\\ u=0 &\mbox{ on } \partial Ω, \end{array} \right. \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $s\in (0,1)$, $N>2s$, $(-{\Delta }_{Dir}{)}^s$ is the fractional Dirichlet Laplacian, $f:\bar{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti-Prodi type assumptions, $t\in \mathbb{R}$, ${\phi }_1$ is the first eigenfunction of the Laplacian with homogenous boundary conditions, and $h:\Omega \to \mathbb{R}$ is a bounded function. Using variational methods, we prove that there exists a $t_0\in \mathbb{R}$ such that the above problem admits at least two distinct solutions for any $t\le t_0$. We also discuss the existence of solutions for a fractional periodic Ambrosetti-Prodi type problem.