In this paper we study the fractional p(., .)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, we prove the existence of weak solutions of corresponding p(., .)-Robin boundary problems with sign-changing potentials by applying variational tools.

中文翻译：

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具有对称变量增长的罗宾分数问题

在本文中，我们研究分数 p(., .)-Laplacian 并为该算子引入相应的非局部共正规导数。我们证明了相应函数空间的基本性质，并为这些算子建立了散度定理的非局部版本。在本文的第二部分，我们通过应用变分工具证明了具有符号变化势的相应 p(., .)-Robin 边界问题的弱解的存在性。