In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractional p&q\documentclass[12pt]{minimal}
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\begin{document}$p\&q$\end{document}-Laplacian equation: 0.1(−Δ)psu+a(x)|u|p−2u+(−Δ)qsu+b(x)|u|q−2u+μ(x)|u|r−2u=λh(x)|u|m−2u,x∈RN,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} &(-\Delta)^{s}_{p}u + a(x)\vert u\vert ^{p-2}u +(-\Delta)^{s}_{q}u + b(x)\vert u\vert ^{q-2}u +\mu(x)\vert u\vert ^{r-2}u \\ &\quad= \lambda h(x)\vert u \vert ^{m-2}u,\quad x\in {\mathbb {R}}^{N}, \end{aligned}$$ \end{document} where λ is a real parameter, (−Δ)ps\documentclass[12pt]{minimal}
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\begin{document}$(-\Delta)_{p}^{s} $\end{document} and (−Δ)qs\documentclass[12pt]{minimal}
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\begin{document}$(-\Delta )_{q}^{s} $\end{document} are the fractional p&q\documentclass[12pt]{minimal}
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\begin{document}$p\&q$\end{document}-Laplacian operators with 0<s<1<q<p,r>1\documentclass[12pt]{minimal}
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\begin{document}$0< s<1<q<p, r>1$\end{document} and sp<N\documentclass[12pt]{minimal}
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\begin{document}$sp< N$\end{document}, and the functions a(x),b(x),μ(x)\documentclass[12pt]{minimal}
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\begin{document}$a(x), b(x),\mu(x)$\end{document}, and h(x)\documentclass[12pt]{minimal}
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\begin{document}$h(x)$\end{document} are nonnegative in RN\documentclass[12pt]{minimal}
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\begin{document}${\mathbb {R}}^{N}$\end{document}. Three cases on p,q,r,m\documentclass[12pt]{minimal}
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\begin{document}$p,q,r,m$\end{document} are considered: p<m<r<ps∗\documentclass[12pt]{minimal}
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\begin{document}$p< m< r< p_{s}^{*}$\end{document}, max{p,r}<m<ps∗\documentclass[12pt]{minimal}
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\begin{document}$\max\{p,r\}< m< p_{s}^{*}$\end{document}, and 1<m<q<r<ps∗\documentclass[12pt]{minimal}
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\begin{document}$1< m< q< r< p_{s}^{*}$\end{document}. Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions to Eq. (0.1) depending on λ,p,q,r,m\documentclass[12pt]{minimal}
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\begin{document}$\lambda, p,q,r,m$\end{document} and the integrability properties of the ratio hr−p/μm−p\documentclass[12pt]{minimal}
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\begin{document}$h^{r-p}/\mu^{m-p}$\end{document}. Our results extend the previous work in Bartolo et al. (J. Math. Anal. Appl. 438:29-41, 2016) and Chaves et al. (Nonlinear Anal. 114:133-141, 2015) to the fractional p&q\documentclass[12pt]{minimal}
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\begin{document}$p\&q$\end{document}-Laplacian equation (0.1).
