An Ambrosetti-Prodi type result for fractional spectral problems

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \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, $f: \bar{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}$ is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti-Prodi type assumptions, $t\in \mathbb{R}$, $\varphi_{1}$ is the first eigenfunction of the Laplacian with homogenous boundary conditions, and $h:\Omega\rightarrow \mathbb{R}$ 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 $t\leq t_{0}$. We also discuss the existence of solutions for a fractional periodic Ambrosetti-Prodi type problem.