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

We consider the following class of non-local superlinear parametric problem { (- increment )su = A.u + u2*+ + f (x), in S2, s-1 u=0, in RN \ S2, where 0 < s < 1, S2 is a bounded domain in RN with N > 2s and 2*s = 2N/(N - 2s) is the fractional critical Sobolev exponent, u+(x) := max{u(x), 0} and A. > 0 is a parameter. When A. is not an eigenvalue of (- increment )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, A. = A.1,s with A.1,s is the principal eigenvalue of (- increment )s) via Ekeland variational principle. (c) 2023 Elsevier B.V. All rights reserved.