We consider the nonlinear fractional problem$$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$We show that ground state solutions converge (along a subsequence) in \(L^2_{\mathrm {loc}} (\mathbb {R}^N)\), under suitable conditions on f and V, to a ground state solution of the local problem as \(s \rightarrow 1^-\).

中文翻译：

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$$ \ mathbb {R} ^ N $$ R N上分数阶Schrödinger方程基态的非局部到局部跃迁

我们考虑非线性分数问题$$ \ begin {aligned}（-\ Delta）^ {s} u + V（x）u = f（x，u）＆\ quad \ hbox {in} \ mathbb {R} ^ N \ end {aligned} $$我们证明了在f的合适条件下，基态解在子序列中收敛于\（L ^ 2 _ {\ mathrm {loc}}（\ mathbb {R} ^ N）\）和V到\（s \ rightarrow 1 ^-\）的局部问题的基态解。