Existence and multiplicity results for a critical superlinear fractional Ambrosetti–Prodi type problem

Abstract

We consider the following class of non-local superlinear parametric problem (−Δ)su=λu+u+2s∗−1+f(x),inΩ,u=0,inRN∖Ω,where 0<s<1, Ω is a bounded domain in RN with N>2s and 2s∗=2N/(N−2s) is the fractional critical Sobolev exponent, u+(x)≔max{u(x),0} and λ>0 is a parameter.When λ is not an eigenvalue of (−Δ)s and N>6s, we apply variational methods (especially Linking Theorem) to show that the above problem has at least two non-trivial solutions. We also discuss the existence results of resonant problem (that is, λ=λ1,s with λ1,s is the principal eigenvalue of (−Δ)s) via Ekeland variational principle.