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Non-local to local transition for ground states of fractional Schrödinger equations on bounded domains. (English) Zbl 1477.35231

Summary: We show that ground state solutions to the nonlinear, fractional problem \[ \begin{cases} (-\Delta)^s u+V(x)u = f(x,u) & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb{R}^N \setminus \Omega, \end{cases} \] on a bounded domain \(\Omega \subset \mathbb{R}^N\), converge (along a subsequence) in \(L^2 (\Omega)\), under suitable conditions on \(f\) and \(V\), to a solution of the local problem as \(s \to 1^-\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35R01 PDEs on manifolds
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