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 \(s \to 1^-\).


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
Full Text: DOI arXiv


[1] O.G. Bakunin, Turbulence and Diffusion: Scaling Versus Equations, Springer, Berlin, 2008. · Zbl 1172.76001
[2] U. Biccari and V. Hernández-Santamaría, The Poisson equation from non-local to local, Electron. J. Differential Equations 2018 (2018), no. 145, 1-13. · Zbl 1396.35066
[3] B. Bieganowski and J. Mederski, Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities, Commun. Pure Appl. Anal. 17 (2018), 143-161. · Zbl 1375.35475
[4] J.P. Borthagaray and P. Ciarlet Jr., On the convergence in the \(H^1\)-norm for the fractional Laplacian, SIAM J. Numer. Anal. 57 (2019), Issue 4, 1723-1743. · Zbl 1422.65371
[5] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, pp. 439-455. · Zbl 1103.46310
[6] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), 225-236. · Zbl 1084.26009
[7] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023
[8] S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Comm. Math. Phys. 333 (2015), no. 2, 1061-1105. · Zbl 1311.35313
[9] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028. · Zbl 1181.35006
[10] G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and Its Applications, vol. 162, Cambridge University Press, 2016. · Zbl 1356.49003
[11] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), no. 12, 3802-3822. · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013
[12] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications (David Yang Gao and Dumitru Motreanu, eds.), International Press, Boston, 2010, pp. 597-632. · Zbl 1218.58010
[13] J.L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations (Oslo, 2010), Abel Symp., vol. 7, Springer, Heidelberg, 2012, pp. 271-298. · Zbl 1248.35223
[14] M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal. 42 (2015), no. 2, 499-547. · Zbl 1307.31022
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