In this article, the problem to be studied is the following [graphic not available: see fulltext]where Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is a bounded regular domain in RN(N≥2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N(N\ge 2)$$\end{document} containing the origin, p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s\in (0,1)$$\end{document}, (N>ps)\documentclass[12pt]{minimal}
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\begin{document}$$(N>ps)$$\end{document}, 0≤β<1/cH\documentclass[12pt]{minimal}
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\begin{document}$$0\le \beta <1/c_{H}$$\end{document} where cH\documentclass[12pt]{minimal}
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\begin{document}$$c_{H}$$\end{document} is the best constant in the fractional Hardy inequality, λ>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0$$\end{document}, f:Ω×R⟶R\documentclass[12pt]{minimal}
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\begin{document}$$f : \Omega \times \mathbb {R} \longrightarrow \mathbb {R}$$\end{document} is a discontinuous function with respect to u and (-Δ)ps\documentclass[12pt]{minimal}
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\begin{document}$$(-\Delta )_p^s$$\end{document} is the fractional p-Laplacian defined as (-Δ)psu(x):=2limε→0∫RN\Bε(x)|u(x)-u(y)|p-2(u(x)-u(y))|x-y|N+spdy,x∈RN,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (-\Delta )_{p}^{s} u(x) :=\displaystyle 2 \lim _{\varepsilon \rightarrow 0} \int _{\mathbb {R}^N \setminus B_{\varepsilon }(x)} \dfrac{\vert u(x)-u(y) \vert ^{p-2}(u(x)-u(y))}{\vert x-y \vert ^{N+sp}} ~dy, ~~~~ x \in \mathbb {R}^N, \end{aligned}$$\end{document}where Bε(x)\documentclass[12pt]{minimal}
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\begin{document}$$B_{\varepsilon }(x)$$\end{document} is the open ε\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon $$\end{document}-ball of centre x and radius ε\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon $$\end{document}. We show that the problem (P) admits at at least one nontrivial weak solution.
