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\\mathcal{K}\\left(\\int_{\\mathbb{R}^{2N}}\\frac{1}{p(x,y)}\\frac{|\\varphi(x)-\\varphi(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \\,dx\\,dy\\right)(-\\Delta)^{s(\\cdot)}_{p(\\cdot)}\\varphi(x) =f(x,\\varphi)\n  \\quad \\mbox{in }\\Omega,\n  \\\\ \\varphi=0 \\quad \\mbox{on }\\mathbb{R}^N\\backslash\\Omega. \\end{array} \\right. \\end{eqnarray*} Under some suitable conditions on the functions $p,s, \\mathcal{K}$ and $f$, the existence and multiplicity of nontrivial 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