# American Institute of Mathematical Sciences

July 2019, 39(7): 4279-4302. doi: 10.3934/dcds.2019173

## The semirelativistic Choquard equation with a local nonlinear term

 1 Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland 2 Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, I-20125 Milano, Italy

* Corresponding author

Received  November 2018 Revised  January 2019 Published  April 2019

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in
 $\mathbb{R}^N$
 $\begin{equation*} \sqrt{ -\Delta + m^2} u - mu + V(x)u = \left( \int_{ \mathbb{R} ^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*}$
where
 $m > 0$
and the potential
 $V$
is decomposed as the sum of a
 $\mathbb{Z}^N$
-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space
 $\mathbb{R}_{+}^{N+1}$
.
Citation: Bartosz Bieganowski, Simone Secchi. The semirelativistic Choquard equation with a local nonlinear term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4279-4302. doi: 10.3934/dcds.2019173
##### References:
 [1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y. [2] B. Bieganowski, Solutions of the fractional Schródinger equation with a sign-changing nonlinearity, J. Math. Anal. Appl., 450 (2017), 461-479. doi: 10.1016/j.jmaa.2017.01.037. [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. doi: 10.3934/cpaa.2018009. [4] V. I. Bogachev, Measure Theory, Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [5] X. Cabré and J. Solà-Morales, Layers solutions in a half-space for boundary reactions, Comm. Pure Applied Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. [6] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [8] Y. H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842. doi: 10.1088/0951-7715/29/6/1827. [9] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. [10] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8. [11] S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schródinger-Newton system, Discrete Continuous Dynmical Systems Series S, 6 (2013), 891-908. doi: 10.3934/dcdss.2013.6.891. [12] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schródinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 (2010), 973-1009. doi: 10.1017/S0308210509000584. [13] S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 73-90. doi: 10.1017/S0308210513000450. [14] V. Coti Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R} ^n}$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [15] V. Coti Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudorelativistic Schródinger equations, Red. Lincei Mat. Appl., 22 (2011), 51-72. doi: 10.4171/RLM/587. [16] A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134. [17] J. Fröhlich and E. Lenzmann, Mean-field Limit of Quantum Bose Gases and Nonlinear Hartree Equation, in Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, 26 pp., École Polytech., Palaiseau, 2004. [18] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schródinger equation of ${\mathbb{R} ^N}$, Indiana Univ. Math. Journal, 54 (2005), 443-464. doi: 10.1512/iumj.2005.54.2502. [19] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. [20] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001. doi: 10.1090/gsm/014. [21] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. [22] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. IHP, Analyse Non Linéaire, 1 (1984), 109-145. [23] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [24] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schródinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733–2742. doi: 10.1088/0264-9381/15/9/019. [25] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007. [26] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256. [27] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. [28] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R} ^N}$, Journal of Mathematical Physics, 54 (2013), 031501, 17 pp. doi: 10.1063/1.4793990. [29] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3, Springer, Berlin; UMI, Bologna, 2007. [30] P. Tod, The ground state energy of the Schródinger-Newton equation, Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7. [31] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.

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##### References:
 [1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y. [2] B. Bieganowski, Solutions of the fractional Schródinger equation with a sign-changing nonlinearity, J. Math. Anal. Appl., 450 (2017), 461-479. doi: 10.1016/j.jmaa.2017.01.037. [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. doi: 10.3934/cpaa.2018009. [4] V. I. Bogachev, Measure Theory, Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [5] X. Cabré and J. Solà-Morales, Layers solutions in a half-space for boundary reactions, Comm. Pure Applied Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. [6] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [8] Y. H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842. doi: 10.1088/0951-7715/29/6/1827. [9] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688. [10] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8. [11] S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schródinger-Newton system, Discrete Continuous Dynmical Systems Series S, 6 (2013), 891-908. doi: 10.3934/dcdss.2013.6.891. [12] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schródinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 (2010), 973-1009. doi: 10.1017/S0308210509000584. [13] S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 73-90. doi: 10.1017/S0308210513000450. [14] V. Coti Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R} ^n}$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [15] V. Coti Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudorelativistic Schródinger equations, Red. Lincei Mat. Appl., 22 (2011), 51-72. doi: 10.4171/RLM/587. [16] A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134. [17] J. Fröhlich and E. Lenzmann, Mean-field Limit of Quantum Bose Gases and Nonlinear Hartree Equation, in Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, 26 pp., École Polytech., Palaiseau, 2004. [18] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schródinger equation of ${\mathbb{R} ^N}$, Indiana Univ. Math. Journal, 54 (2005), 443-464. doi: 10.1512/iumj.2005.54.2502. [19] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. [20] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001. doi: 10.1090/gsm/014. [21] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. [22] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. IHP, Analyse Non Linéaire, 1 (1984), 109-145. [23] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [24] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schródinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733–2742. doi: 10.1088/0264-9381/15/9/019. [25] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007. [26] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256. [27] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. [28] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R} ^N}$, Journal of Mathematical Physics, 54 (2013), 031501, 17 pp. doi: 10.1063/1.4793990. [29] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3, Springer, Berlin; UMI, Bologna, 2007. [30] P. Tod, The ground state energy of the Schródinger-Newton equation, Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7. [31] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.
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