An Ambrosetti–Prodi type result for fractional spectral problems

We consider the following class of fractional parametric problems( − Δ Dir ) s u = f ( x , u ) + t φ 1 + hin Ω ,u = 0on ∂ Ω ,where Ω ⊂ R Nis a smooth bounded domain, s ∈ ( 0 , 1 ) , N > 2 s ,( − Δ Dir ) s is the fractional Dirichlet Laplacian, f : Ω ¯ × R → R is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, t ∈ R , φ 1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and h : Ω → R is a bounded function. Using variational methods, we prove that there exists at 0 ∈ R such that the above problem admits at least two distinct solutions for any t ≤ t 0 . We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.