In this paper, we consider a class of fractional Laplacian problems of the form: $\begin{array}{rl}& (-\mathrm{\Delta }{)}_{p_1(x,.)}^{s}u+(-\mathrm{\Delta }{)}_{p_2(x,.)}^{s}u+|u{|}^{q\left(x\right)-2}u=g\left(x\right)u^{-\gamma \left(x\right)}\\ & +\lambda f(x,u)\text{in }\mathrm{\Omega },u=0,\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbf{R}}^N,(N\ge 2),$ is a bounded domain and $(-\mathrm{\Delta }{)}_{p_i(x,.)}^s$ is the fractional $p_i(x,.)$-Laplacian. We assume that λ is a nonnegative parameter and $\gamma :\overline{\mathrm{\Omega }}\to (0,1)$ is a continuous function. By using variational methods and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces, we prove the existence of solutions to problem $\left(P_{\lambda }\right)$.

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具有可变指数和不定权重的奇异分数式拉普拉斯问题的解的存在性

在本文中，我们考虑以下形式的一类分数拉普拉斯问题： $\begin{array}{rl}& (-\mathrm{\Delta }{)}_{{磷}_1(X,.)}^{秒}你+(-\mathrm{\Delta }{)}_{{磷}_2(X,.)}^{秒}你+|你{|}^{q\left(X\right)-2}你=G\left(X\right){你}^{-\gamma \left(X\right)}\\ & +\lambda F(X,你)\text{在 }\mathrm{\Omega },你=0,\text{在}\mathrm{\partial }\mathrm{\Omega },\end{array}$ 在哪里 $\mathrm{\Omega }\subset {\mathbf{电阻}}^N,(N\ge 2),$ 是一个有界域并且 $(-\mathrm{\Delta }{)}_{{磷}_{\mathrm{一世}}(X,.)}^{秒}$ 是小数 ${磷}_{\mathrm{一世}}(X,.)$- 拉普拉斯。我们假设λ是一个非负参数，并且$\gamma ：\overline{\mathrm{\Omega }}\to (0,1)$是连续函数。通过使用变分方法和单调性论证，结合广义 Lebesgue Sobolev 空间的理论，我们证明了问题的解的存在性$\left({磷}_{\lambda }\right)$.