We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.

中文翻译：

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连接 Neumann 分数 p-拉普拉斯算子的锥体

我们考虑在存在非局部 Neumann 边界条件的情况下由分数 $p-$Laplacian 控制的非线性问题。我们面临两个问题。第一：$p-$superlinear 项可能不满足 Ambrosetti-Rabinowitz 条件。其次，也是更重要的一点：尽管底层泛函的拓扑结构提醒了链接定理之一，但相关的本征函数的非局部性质阻止了这种经典定理的使用。由于这些原因，我们被引导采用另一种方法，依赖于通过锥体连接的概念。