The critical fractional Ambrosetti–Prodi problem

Abstract

AbstractIn this paper we focus on the following nonlocal problem with critical growth: $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u = \lambda u + u_{+}^{2^{*}_{s}-1} + f(x) &{} \text{ in } \Omega ,\\ u=0 &{} \text{ in } \mathbb {R}^{N}\setminus \Omega , \end{array} \right. \end{aligned}$$ ( - Δ ) s u = λ u + u + 2 s ∗ - 1 + f ( x ) in Ω , u = 0 in R N \ Ω , where $$s\in (0, 1)$$ s ∈ ( 0 , 1 ) , $$N>2s$$ N > 2 s , $$\Omega \subset \mathbb {R}^{N}$$ Ω ⊂ R N is a smooth bounded domain, $$\lambda >0$$ λ > 0 , $$(-\Delta )^{s}$$ ( - Δ ) s is the fractional Laplacian, $$f= te_{1}+h$$ f = t e 1 + h where $$t\in \mathbb {R}$$ t ∈ R , $$e_{1}$$ e 1 is the first eigenfunction of $$(-\Delta )^{s}$$ ( - Δ ) s with homogeneous Dirichlet boundary datum, and $$h\in L^{\infty }(\Omega )$$ h ∈ L ∞ ( Ω ) is such that $$\int _{\Omega } h e_{1}\, dx=0$$ ∫ Ω h e 1 d x = 0 . According to the interaction of the nonlinear term with the spectrum of $$(-\Delta )^{s}$$ ( - Δ ) s , we establish some existence and multiplicity results for the above problem by means of variational methods.