# American Institute of Mathematical Sciences

December  2019, 39(12): 7113-7139. doi: 10.3934/dcds.2019298

## The fractional Schrödinger equation with singular potential and measure data

Dedicated to Luis Caffarelli with deep appreciation.

Received  December 2018 Revised  April 2019 Published  September 2019

We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of $(-\Delta)^s$ and prove well-posedness for functions as data. If $V$ is bounded or mildly singular a unique solution of $(-\Delta)^s u + V u = \mu$ exists for every Borel measure $\mu$. On the other hand, when $V$ is allowed to be more singular, but only on a finite set of points, a solution of $(-\Delta)^s u + V u = \delta_x$, where $\delta_x$ is the Dirac measure at $x$, exists if and only if $h(y) = V(y) |x - y|^{-(n+2s)}$ is integrable on some small ball around $x$. We prove that the set $Z = \{x \in \Omega : \rm{no solution of } (-\Delta)^s u + Vu = \delta_x \rm{ exists}\}$ is relevant in the following sense: a solution of $(-\Delta)^s u + V u = \mu$ exists if and only if $|\mu| (Z) = 0$. Furthermore, $Z$ is the set points where the strong maximum principle fails, in the sense that for any bounded $f$ the solution of $(-\Delta)^s u + Vu = f$ vanishes on $Z$.

Citation: David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298
##### References:
 [1] P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, Journal of Evolution Equations, 3 (2003), 673-770.  doi: 10.1007/s00028-003-0117-8. [2] P. Bénilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^1(R^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555. [3] M. Bonforte, A. Figalli and J. Vázquez, Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations, Calculus of Variations and Partial Differential Equations, 57 (2018), Art. 57, 34 pp. doi: 10.1007/s00526-018-1321-2. [4] M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete and Continuous Dynamical Systems- Series A, 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725. [5] M. Bonforte and J. L. Vázquez, Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds, Nonlinear Analysis, Theory, Methods and Applications, 131 (2016), 363-398.  doi: 10.1016/j.na.2015.10.005. [6] H. Brezis, M. Marcus and A. C. Ponce, A new concept of reduced measure for nonlinear elliptic equations, Comptes Rendus Mathematique, 339 (2004), 169-174.  doi: 10.1016/j.crma.2004.05.012. [7] H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, in Mathematical Aspects of Nonlinear Dispersive Equations (eds. J. Bourgain, C. Kenig and S. Klainerman), vol. 163 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2007, 55-109. [8] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [9] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [10] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [11] L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 33 (2016), 767-807. [12] H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, Journal of Differential Equations, 257 (2014), 1457-1486.  doi: 10.1016/j.jde.2014.05.012. [13] M. Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Annali di Matematica Pura ed Applicata, 196 (2017), 555-578.  doi: 10.1007/s10231-016-0586-3. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathematiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [15] J. I. Díaz, D. Gómez-Castro and J. Vázquez, The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Analysis, 177 (2018), 325-360. [16] J. I. Díaz, D. Gómez-Castro, J.-M. Rakotoson and R. Temam, Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach, Discrete and Continuous Dynamical Systems, 38 (2018), 509-546.  doi: 10.3934/dcds.2018023. [17] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019. [18] M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3. [19] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [20] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018. [21] K.-Y. Kim and P. Kim, Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1, \eta}$ open sets, Stochastic Process. Appl., 124 (2014), 3055-3083.  doi: 10.1016/j.spa.2014.04.004. [22] T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, in Communications in Mathematical Physics, 337 (2015), 1317–1368. doi: 10.1007/s00220-015-2356-2. [23] L. Orsina and A. C. Ponce, On the nonexistence of Green's function and failure of the strong maximum principle, Journal de Mathématiques Pures et Appliquées, 2019. doi: 10.1016/j.matpur.2019.06.001. [24] A. C. Ponce, Elliptic PDEs, Measures and Capacities From the Poisson Equation to Nonlinear Thomas-Fermi Problems, European Mathematical Society Publishing House, Zurich, 2016. doi: 10.4171/140. [25] A. C. Ponce and N. Wilmet, Schrödinger operators involving singular potentials and measure data, Journal of Differential Equations, 263 (2017), 3581-3610.  doi: 10.1016/j.jde.2017.04.039. [26] J. M. Rakotoson, Potential capacity and applications, 2018, URL http://arXiv.org/abs/1812.04061, arXiv preprint. [27] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publicacions Matemàtiques, 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01. [28] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal des Mathematiques Pures et Appliquees, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [29] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. [30] J. L. Vázquez, On a Semilinear Equation in $\mathbb R^2$ Involving Bounded Measures, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 95 (1983), 181-202.  doi: 10.1017/S0308210500012907.

show all references

Dedicated to Luis Caffarelli with deep appreciation.

##### References:
 [1] P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, Journal of Evolution Equations, 3 (2003), 673-770.  doi: 10.1007/s00028-003-0117-8. [2] P. Bénilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^1(R^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555. [3] M. Bonforte, A. Figalli and J. Vázquez, Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations, Calculus of Variations and Partial Differential Equations, 57 (2018), Art. 57, 34 pp. doi: 10.1007/s00526-018-1321-2. [4] M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete and Continuous Dynamical Systems- Series A, 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725. [5] M. Bonforte and J. L. Vázquez, Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds, Nonlinear Analysis, Theory, Methods and Applications, 131 (2016), 363-398.  doi: 10.1016/j.na.2015.10.005. [6] H. Brezis, M. Marcus and A. C. Ponce, A new concept of reduced measure for nonlinear elliptic equations, Comptes Rendus Mathematique, 339 (2004), 169-174.  doi: 10.1016/j.crma.2004.05.012. [7] H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, in Mathematical Aspects of Nonlinear Dispersive Equations (eds. J. Bourgain, C. Kenig and S. Klainerman), vol. 163 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2007, 55-109. [8] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [9] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [10] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [11] L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 33 (2016), 767-807. [12] H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, Journal of Differential Equations, 257 (2014), 1457-1486.  doi: 10.1016/j.jde.2014.05.012. [13] M. Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Annali di Matematica Pura ed Applicata, 196 (2017), 555-578.  doi: 10.1007/s10231-016-0586-3. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathematiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [15] J. I. Díaz, D. Gómez-Castro and J. Vázquez, The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Analysis, 177 (2018), 325-360. [16] J. I. Díaz, D. Gómez-Castro, J.-M. Rakotoson and R. Temam, Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach, Discrete and Continuous Dynamical Systems, 38 (2018), 509-546.  doi: 10.3934/dcds.2018023. [17] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019. [18] M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3. [19] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [20] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018. [21] K.-Y. Kim and P. Kim, Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1, \eta}$ open sets, Stochastic Process. Appl., 124 (2014), 3055-3083.  doi: 10.1016/j.spa.2014.04.004. [22] T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, in Communications in Mathematical Physics, 337 (2015), 1317–1368. doi: 10.1007/s00220-015-2356-2. [23] L. Orsina and A. C. Ponce, On the nonexistence of Green's function and failure of the strong maximum principle, Journal de Mathématiques Pures et Appliquées, 2019. doi: 10.1016/j.matpur.2019.06.001. [24] A. C. Ponce, Elliptic PDEs, Measures and Capacities From the Poisson Equation to Nonlinear Thomas-Fermi Problems, European Mathematical Society Publishing House, Zurich, 2016. doi: 10.4171/140. [25] A. C. Ponce and N. Wilmet, Schrödinger operators involving singular potentials and measure data, Journal of Differential Equations, 263 (2017), 3581-3610.  doi: 10.1016/j.jde.2017.04.039. [26] J. M. Rakotoson, Potential capacity and applications, 2018, URL http://arXiv.org/abs/1812.04061, arXiv preprint. [27] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publicacions Matemàtiques, 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01. [28] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal des Mathematiques Pures et Appliquees, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [29] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. [30] J. L. Vázquez, On a Semilinear Equation in $\mathbb R^2$ Involving Bounded Measures, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 95 (1983), 181-202.  doi: 10.1017/S0308210500012907.
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