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Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth. (English) Zbl 1418.35168

Summary: We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy-Littlewood-Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti-Rabinowitz-type conditions, still using variational methods.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
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[1] N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math. 7 (2005), no. 3, 269-298. · Zbl 1070.35083
[2] C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in {\mathbb{R}^{2}}, J. Differential Equations 261 (2016), no. 3, 1933-1972. · Zbl 1347.35096
[3] C. O. Alves, J. A. M. do Ó and M. A. S. Souto, Local mountain-pass for a class of elliptic problems in {\mathbb{R}^{N}} involving critical growth, Nonlinear Anal. 46 (2001), no. 4, 495-510. · Zbl 1113.35323
[4] C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations 263 (2017), no. 7, 3943-3988. · Zbl 1378.35113
[5] C. O. Alves, M. Squassina and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23-58. · Zbl 1366.35050
[6] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345. · Zbl 0533.35029
[7] C. Bonanno, P. d’Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl. 417 (2014), no. 1, 180-199. · Zbl 1332.35066
[8] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477. · Zbl 0541.35029
[9] J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc. 362 (2010), no. 4, 1981-2001. · Zbl 1188.35082
[10] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185-200. · Zbl 1132.35078
[11] J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1859-1899. · Zbl 1294.35131
[12] J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc. 229 (2014), no. 1076, 1-89. · Zbl 1303.35094
[13] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II, Calc. Var. Partial Differential Equations 18 (2003), no. 2, 207-219. · Zbl 1073.35199
[14] J. Byeon, J. Zhang and W. Zou, Singularly perturbed nonlinear Dirichlet problems involving critical growth, Calc. Var. Partial Differential Equations 47 (2013), no. 1-2, 65-85. · Zbl 1270.35042
[15] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343. · Zbl 1093.45001
[16] S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 5, 973-1009. · Zbl 1215.35146
[17] P. d’Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal. 262 (2012), no. 10, 4600-4633. · Zbl 1239.35152
[18] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121-137. · Zbl 0844.35032
[19] M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 2, 127-149. · Zbl 0901.35023
[20] M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), no. 3, 883-898. · Zbl 0932.35080
[21] M. D. Donsker and S. R. S. Varadhan, The polaron problem and large deviations, Phys. Rep. 77 (1981), no. 3, 235-237.
[22] M. D. Donsker and S. R. S. Varadhan, Asymptotics for the polaron, Comm. Pure Appl. Math. 36 (1983), no. 4, 505-528. · Zbl 0538.60081
[23] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), no. 3, 397-408. · Zbl 0613.35076
[24] H. Fröhlich, Theory of electrical breakdown in ionic crystal, Proc. Roy. Soc. Ser. A 160 (1937), 230-241.
[25] F. Gao and M. Yang, Existence and multiplicity of solutions for a class of Choquard equations with Hardy-Littlewood-Sobolev critical exponent, preprint (2016), .
[26] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), no. 2, 1006-1041. · Zbl 1357.35106
[27] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math. (2017), 10.1007/s11425-000-0000-0.
[28] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in {{\mathbf{R}}^{n}}, Mathematical Analysis and Applications. Part A, Adv. in Math. Suppl. Stud. 7, Academic Press, New York (1981), 369-402.
[29] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. · Zbl 0562.35001
[30] E. P. Gross, Physics of many-Particle Systems. Vol. 1, Gordon Breach, New York, 1996.
[31] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), no. 10, 1633-1659. · Zbl 0877.35091
[32] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on {{\mathbf{R}}^{N}}, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787-809. · Zbl 0935.35044
[33] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on {\mathbb{R}^{N}}, Indiana Univ. Math. J. 54 (2005), no. 2, 443-464. · Zbl 1143.35321
[34] L. Jeanjean and J. F. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 1, 23-28. · Zbl 0996.47052
[35] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1976/77), no. 2, 93-105. · Zbl 0369.35022
[36] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349-374. · Zbl 0527.42011
[37] E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. · Zbl 0966.26002
[38] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), no. 3, 185-194.
[39] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063-1072. · Zbl 0453.47042
[40] P.-L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, Nonlinear Problems: Present and Future (Los Alamos 1981), North-Holland Math. Stud. 61, North-Holland, Amsterdam, (1982), 17-34.
[41] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004
[42] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455-467. · Zbl 1185.35260
[43] M. Macrì and M. Nolasco, Stationary solutions for the non-linear Hartree equation with a slowly varying potential, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 6, 681-715. · Zbl 1203.35264
[44] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153-184. · Zbl 1285.35048
[45] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557-6579. · Zbl 1325.35052
[46] V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), no. 1-2, 199-235. · Zbl 1309.35029
[47] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), no. 1, 773-813. · Zbl 1360.35252
[48] M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal. 9 (2010), no. 5, 1411-1419. · Zbl 1202.35304
[49] Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class {(V)_{a}}, Comm. Partial Differential Equations 13 (1988), no. 12, 1499-1519. · Zbl 0702.35228
[50] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie, Berlin, 1954.
[51] R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28 (1996), no. 5, 581-600. · Zbl 0855.53046
[52] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1743, 1927-1939. · Zbl 1152.81659
[53] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf, New York, 2005. · Zbl 1188.00007
[54] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270-291. · Zbl 0763.35087
[55] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655-674. · Zbl 1136.35037
[56] S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), no. 9-10, 3842-3856. · Zbl 1187.35254
[57] M. Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), no. 1-2, 19-64. · Zbl 0646.53005
[58] X. Sun and Y. Zhang, Multi-peak solution for nonlinear magnetic Choquard type equation, J. Math. Phys. 55 (2014), no. 3, Article ID 031508. · Zbl 1296.35180
[59] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), no. 2, 229-244. · Zbl 0795.35118
[60] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys. 50 (2009), no. 1, Article ID 012905. · Zbl 1189.81061
[61] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.
[62] M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, New York, 2013. · Zbl 1284.46001
[63] M. Yang and Y. Ding, Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Commun. Pure Appl. Anal. 12 (2013), no. 2, 771-783. · Zbl 1270.35218
[64] M. Yang, J. Zhang and Y. Zhang, Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Commun. Pure Appl. Anal. 16 (2017), no. 2, 493-512. · Zbl 1364.35027
[65] J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, J. Lond. Math. Soc. (2) 90 (2014), no. 3, 827-844. · Zbl 1317.35247
[66] J. Zhang, J. A. M. do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity, Commun. Contemp. Math. 19 (2017), no. 4, Article ID 1650028.
[67] J. Zhang and W. Zou, A Berestycki-Lions theorem revisited, Commun. Contemp. Math. 14 (2012), no. 5, Article ID 1250033. · Zbl 1254.35068
[68] J. Zhang and W. Zou, Solutions concentrating around the saddle points of the potential for critical Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 4119-4142. · Zbl 1339.35109
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