Non-local to local transition for ground states of fractional Schrödinger equations on bounded domains

Bartosz Bieganowski, Simone Secchi



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


Variational methods; fractional Schrödinger equation; non-local to local transition; ground state; Nehari manifold

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O.G. Bakunin, Turbulence and Diffusion: Scaling Versus Equations, Springer, Berlin, 2008.

U. Biccari and V. Hernández-Santamarı́a, The Poisson equation from non-local to local, Electron. J. Differential Equations 2018 (2018), no. 145, 1–13.

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.

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.

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, pp. 439–455

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), 225–236.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

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.

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005–1028.

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.

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), no. 12, 3802–3822.

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.

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.

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.


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