In this paper, our goal is to establish a generalized version of Lions-type theorem for the fractional Laplacian. As an application of this theorem, we consider the existence of ground state solutions of a fractional equation:\[(-\Delta)^s u + V (\lvert x \rvert) u = f(u), \; x \in \mathbb{R}^N ,\]where $N \geqslant 3, s \in (\frac{1}{2}, 1), V$ is a singular potential with $\alpha \in (0, 2s) \cup (2s, 2N - 2s)$, and the nonlinearity $f$ has the critical growth, discussed without any boundary value condition.

中文翻译：

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分数拉普拉斯算子的狮子型定理及其应用

在本文中，我们的目标是为分数拉普拉斯算子建立一个广义版本的 Lions 型定理。作为该定理的应用，我们考虑分数方程的基态解的存在性：\[(-\Delta)^su + V (\lvert x \rvert) u = f(u), \; x \in \mathbb{R}^N ,\]where $N \geqslant 3, s \in (\frac{1}{2}, 1), V$ 是奇异势，$\alpha \in (0 , 2s) \cup (2s, 2N - 2s)$，非线性$f$有临界增长，没有任何边界值条件讨论。