# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021061

## Multiple positive bound state solutions for a critical Choquard equation

 1 Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB, Brazil 2 Departamento de Matemática, Universidade de Brasilia - UNB, 70910-900, Brasília-DF, Brazil 3 Dipartimento di Matematica, Università di Roma "Tor Vergata", CEP: 00133, Roma, Italia

* Corresponding author: R. Molle

Received  August 2020 Published  March 2021

In this paper we consider the problem
 $(P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.$
where
 $V_{\lambda} = \lambda+V_{0}$
with
 $\lambda \geq 0$
,
 $V_0\in L^{N/2}({\mathbb{R}}^N)$
,
 $I_{\mu} = \frac{1}{|x|^\mu}$
is the Riesz potential with
 $0<\mu<\min\{N, 4\}$
and
 $2^{*}_{\mu} = \frac{2N-\mu}{N-2}$
with
 $N\geq 3$
. Under some smallness assumption on
 $V_0$
and
 $\lambda$
we prove the existence of two positive solutions of
 $(P_\lambda)$
. In order to prove the main results, we used variational methods combined with degree theory.
Citation: Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021061
##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Advanced in Nonlinear Analysis, 5 (2016), 331-345.  doi: 10.1515/anona-2015-0123.  Google Scholar [2] C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Eq., 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.  Google Scholar [3] C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.  doi: 10.1016/S0362-546X(01)00887-2.  Google Scholar [4] C. O. Alves and M. Yang, Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 23-58.  doi: 10.1017/S0308210515000311.  Google Scholar [5] C. O. Alves and Y. Jianfu, Existence and regularity of solutions for a Choquard equation with zero mass, Milan J. Math., 86 (2018), 329-342.  doi: 10.1007/s00032-018-0289-x.  Google Scholar [6] T. Aubin, Problemès Isopérimétriques et Sobolev spaces, J. Diff. Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.  Google Scholar [7] T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, arXiv: 2008.07431, to appear on Comm. in PDE, https://doi.org/10.1080/03605302.2021.1893747. doi: 10.1080/03605302.2021.1893747.  Google Scholar [8] V. Benci, C. R. Grisanti and A. M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with $V(\infty) = 0$, Progr. Nonlinear Differential Equations Appl., 66 (2005), 53-65.  doi: 10.1007/3-7643-7401-2_4.  Google Scholar [9] V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u +a(x)u = u^{\frac{N+2}{N-2}}$ in $\mathbb{R}^N,$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar [10] A.K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Analysis, 26 (1996), 823-833.  doi: 10.1016/0362-546X(94)00324-B.  Google Scholar [11] L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230.   Google Scholar [12] G. Cerami and R. Molle, Multiple positive bound states for critical Schrödinger-Poisson systems, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 73, 29 pp. doi: 10.1051/cocv/2018071.  Google Scholar [13] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Eq., 3 (1995), 493-512.  doi: 10.1007/BF01187898.  Google Scholar [14] P. Cherrier, Meilleures constantes dans les inegalites relatives aux espaces de Sobolev, Bull. Sci. Math., 108 (1984), 225-262.   Google Scholar [15] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8.  Google Scholar [16] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.   Google Scholar [17] L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847–5866, arXiv: 1810.11186v1. doi: 10.3934/dcds.2019219.  Google Scholar [18] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, arXiv: 1810.11759v1. Google Scholar [19] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Science China Mathematics, 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.  Google Scholar [20] F. Gao, E. da Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.  Google Scholar [21] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. of Funct. Anal., 271 (2016), 107–135. doi: 10.1016/j.jfa.2016.04.019.  Google Scholar [22] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [23] P. M. Girão, A sharp inequality for Sobolev functions, C. R. Math. Acad. Sci. Paris, 334 (2002), 105-108.  doi: 10.1016/S1631-073X(02)02215-X.  Google Scholar [24] R. Hadiji, R. Molle, D. Passaseo and H. Yazidi, Localization of solutions for nonlinear elliptic problems with critical growth, C. R. Acad. Sci. Paris Sér. I Math., 343 (2006), 725-730.  doi: 10.1016/j.crma.2006.10.018.  Google Scholar [25] S. Lancelotti and R. Molle, Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 8, 28 pp. doi: 10.1007/s00030-019-0611-5.  Google Scholar [26] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar [27] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [28] E. Lieb and M. Loss, Analysis,, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. doi: 10.1090/gsm/014.  Google Scholar [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations., Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.  Google Scholar [30] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I., Rev. Mat. Iberoamericana, Part I, 1 (1985), 145–201, and Part II, 2 (1985), 45–121. doi: 10.4171/RMI/6.  Google Scholar [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455–467. doi: 10.1007/s00205-008-0208-3.  Google Scholar [32] C. Mercuri and M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete & Continuous Dynamical Systems - A, 28 (2010), 469-493.  doi: 10.3934/dcds.2010.28.469.  Google Scholar [33] O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N$ with critical growth, Nonlinear Analysis, 29 (1997), 773-781.  doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar [34] R. Molle and D. Passaseo, Multispike solutions of nonlinear elliptic equations with critical Sobolev exponent, Comm. in PDE., 32 (2007), 797-818.  doi: 10.1080/03605300600781642.  Google Scholar [35] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar [36] V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equation: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp, arXiv: 1403.7414v1. doi: 10.1142/S0219199715500054.  Google Scholar [37] V. Moroz and J. Van Schaftingen, Ground states 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.  Google Scholar [38] V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations., Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar [39] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar [40] D. Passaseo, Some sufficient conditions for the existence of positive solutions to the equation $-\Delta u +a(x)u = u^{\frac{N+2}{N-2}}$ in bounded domains, Ann. Inst. Henri Poincaré, 13 (1996), 185-227.  doi: 10.1016/S0294-1449(16)30102-0.  Google Scholar [41] S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842–3856. doi: 10.1016/j.na.2010.01.021.  Google Scholar [42] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonliarities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar [43] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Springer-Verlag, Berlin, 2008.  Google Scholar [44] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

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##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Advanced in Nonlinear Analysis, 5 (2016), 331-345.  doi: 10.1515/anona-2015-0123.  Google Scholar [2] C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Eq., 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.  Google Scholar [3] C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.  doi: 10.1016/S0362-546X(01)00887-2.  Google Scholar [4] C. O. Alves and M. Yang, Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 23-58.  doi: 10.1017/S0308210515000311.  Google Scholar [5] C. O. Alves and Y. Jianfu, Existence and regularity of solutions for a Choquard equation with zero mass, Milan J. Math., 86 (2018), 329-342.  doi: 10.1007/s00032-018-0289-x.  Google Scholar [6] T. Aubin, Problemès Isopérimétriques et Sobolev spaces, J. Diff. Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.  Google Scholar [7] T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, arXiv: 2008.07431, to appear on Comm. in PDE, https://doi.org/10.1080/03605302.2021.1893747. doi: 10.1080/03605302.2021.1893747.  Google Scholar [8] V. Benci, C. R. Grisanti and A. M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with $V(\infty) = 0$, Progr. Nonlinear Differential Equations Appl., 66 (2005), 53-65.  doi: 10.1007/3-7643-7401-2_4.  Google Scholar [9] V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u +a(x)u = u^{\frac{N+2}{N-2}}$ in $\mathbb{R}^N,$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar [10] A.K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Analysis, 26 (1996), 823-833.  doi: 10.1016/0362-546X(94)00324-B.  Google Scholar [11] L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230.   Google Scholar [12] G. Cerami and R. Molle, Multiple positive bound states for critical Schrödinger-Poisson systems, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 73, 29 pp. doi: 10.1051/cocv/2018071.  Google Scholar [13] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Eq., 3 (1995), 493-512.  doi: 10.1007/BF01187898.  Google Scholar [14] P. Cherrier, Meilleures constantes dans les inegalites relatives aux espaces de Sobolev, Bull. Sci. Math., 108 (1984), 225-262.   Google Scholar [15] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8.  Google Scholar [16] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.   Google Scholar [17] L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847–5866, arXiv: 1810.11186v1. doi: 10.3934/dcds.2019219.  Google Scholar [18] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, arXiv: 1810.11759v1. Google Scholar [19] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Science China Mathematics, 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.  Google Scholar [20] F. Gao, E. da Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.  Google Scholar [21] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. of Funct. Anal., 271 (2016), 107–135. doi: 10.1016/j.jfa.2016.04.019.  Google Scholar [22] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [23] P. M. Girão, A sharp inequality for Sobolev functions, C. R. Math. Acad. Sci. Paris, 334 (2002), 105-108.  doi: 10.1016/S1631-073X(02)02215-X.  Google Scholar [24] R. Hadiji, R. Molle, D. Passaseo and H. Yazidi, Localization of solutions for nonlinear elliptic problems with critical growth, C. R. Acad. Sci. Paris Sér. I Math., 343 (2006), 725-730.  doi: 10.1016/j.crma.2006.10.018.  Google Scholar [25] S. Lancelotti and R. Molle, Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 8, 28 pp. doi: 10.1007/s00030-019-0611-5.  Google Scholar [26] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar [27] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [28] E. Lieb and M. Loss, Analysis,, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. doi: 10.1090/gsm/014.  Google Scholar [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations., Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.  Google Scholar [30] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I., Rev. Mat. Iberoamericana, Part I, 1 (1985), 145–201, and Part II, 2 (1985), 45–121. doi: 10.4171/RMI/6.  Google Scholar [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455–467. doi: 10.1007/s00205-008-0208-3.  Google Scholar [32] C. Mercuri and M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete & Continuous Dynamical Systems - A, 28 (2010), 469-493.  doi: 10.3934/dcds.2010.28.469.  Google Scholar [33] O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N$ with critical growth, Nonlinear Analysis, 29 (1997), 773-781.  doi: 10.1016/S0362-546X(96)00087-9.  Google Scholar [34] R. Molle and D. Passaseo, Multispike solutions of nonlinear elliptic equations with critical Sobolev exponent, Comm. in PDE., 32 (2007), 797-818.  doi: 10.1080/03605300600781642.  Google Scholar [35] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar [36] V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equation: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp, arXiv: 1403.7414v1. doi: 10.1142/S0219199715500054.  Google Scholar [37] V. Moroz and J. Van Schaftingen, Ground states 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.  Google Scholar [38] V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations., Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar [39] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar [40] D. Passaseo, Some sufficient conditions for the existence of positive solutions to the equation $-\Delta u +a(x)u = u^{\frac{N+2}{N-2}}$ in bounded domains, Ann. Inst. Henri Poincaré, 13 (1996), 185-227.  doi: 10.1016/S0294-1449(16)30102-0.  Google Scholar [41] S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842–3856. doi: 10.1016/j.na.2010.01.021.  Google Scholar [42] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonliarities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar [43] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Springer-Verlag, Berlin, 2008.  Google Scholar [44] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar
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