In this paper we deal with the existence and non-existence of solutions for the following Ambrosetti-Prodi type problem P(-Δ)su=P(x)(g(u)+f(x))inRN,u∈Ds(RN),lim|x|→+∞u(x)=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{lcl} (-\Delta )^s u=P(x)\Big ( g(u)+f(x)\Big ) \text{ in } \mathbb {R}^N,\\ u \in D^{s}(\mathbb R^N),\ \lim _{|x|\rightarrow +\infty }u(x)=0, \end{array} \right. \end{aligned}$$\end{document}where N>2s\documentclass[12pt]{minimal}
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\begin{document}$$N > 2s$$\end{document}, s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s \in (0,1)$$\end{document}, P∈C(RN,R+)\documentclass[12pt]{minimal}
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\begin{document}$$P\in C(\mathbb R^N,\mathbb R^+)$$\end{document}, f∈C1,σ(RN)∩L∞(RN)\documentclass[12pt]{minimal}
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\begin{document}$$f\in C^{1,\sigma }(\mathbb R^N)\cap L^{\infty }(\mathbb R^N)$$\end{document}, g∈C1,σ(R)\documentclass[12pt]{minimal}
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\begin{document}$$g\in C^{1,\sigma }(\mathbb R)$$\end{document} and (-Δ)su\documentclass[12pt]{minimal}
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\begin{document}$$(-\Delta )^su$$\end{document} is the fractional
Laplacian. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.
