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Existence and asymptotic behavior of ground states for Choquard-Pekar equations with Hardy potential and critical reaction. (English) Zbl 1485.35345

This article discusses the existence and asymptotic behavior of ground state solutions for a nonautonomous Choquard-Pekar equation \[ -\Delta u+\Big(K(x)-\frac{\mu}{|x|^2|}\Big) u=(W\ast F(u))f(u)+|u|^{2^*-2}u\quad\mbox{ in }\mathbb{R^N}\setminus\{0\}. \] Where \(N\geq 3\), \(\mu<(N-2)^2/4\), \(2^*=2N/(N-2)\) and \(K\) is allowed to change sign. Under some structural conditions on \(W\) the authors obtain the existence of ground state solutions. It is further shown that as \(\mu\to 0^+\) these ground state solutions converge (up to a \(\mathbb{Z}^N\)-translation) to a ground state solution of the limiting problem.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
46N50 Applications of functional analysis in quantum physics
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[1] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443 (2004) · Zbl 1059.35037
[2] Ackermann, N., A Cauchy-Schwarz type inequality for bilinear integrals on positive measures, Proc. Am. Math. Soc., 133, 2647-2656 (2005) · Zbl 1066.26013
[3] Albanese, C., Localised solutions of Hartree equations for narrow-band crystals, Commun. Math. Phys., 120, 97-103 (1988) · Zbl 0671.35071
[4] Bartsch, T.; Ding, YH, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313, 15-37 (1999) · Zbl 0927.35103
[5] Buffoni, B.; Jeanjean, L.; Stuart, CA, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Am. Math. Soc., 119, 179-186 (1993) · Zbl 0789.35052
[6] Cao, D.; Peng, S., A note on the sign-changing solutions to elliptic problem with critical Sobolev and Hardy terms, J. Differ. Equ., 193, 424-434 (2003) · Zbl 1140.35412
[7] Catto, I.; Le Bris, C.; Lions, PL, On some periodic Hartree-type models for crystals, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 19, 143-190 (2002) · Zbl 1005.81101
[8] Chen, Y.; Tang, XH, Nehari-type ground state solutions for Schrödinger equations with Hardy potential and critical nonlinearities, Complex Var. Elliptic Equ., 65, 1315-1335 (2020) · Zbl 1454.35071
[9] Deng, Y.; Jin, L.; Peng, S., Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differ. Equ., 253, 1376-1398 (2012) · Zbl 1248.35058
[10] Fröhlich, H., Theory of electrical breakdown in ionic crystal, Proc. R. Soc. Edinb. Sect. A, 160, 901, 230-241 (1937)
[11] Ghergu, M.; Taliaferro, S., Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differ. Equ., 261, 189-217 (2016) · Zbl 1382.35113
[12] Ghimenti, M.; Van Schaftingen, J., Nodal solutions for the Choquard equation, J. Funct. Anal., 271, 107-135 (2016) · Zbl 1345.35046
[13] Giulini, D.; Großardt, A., The Schrödinger-Newton equation as a non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields, Class. Quantum Gravity, 29, 21, 215010 (2012) · Zbl 1266.83009
[14] Guo, Q.; Mederski, J., Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potential, J. Differ. Equ., 260, 4180-4202 (2016) · Zbl 1335.35232
[15] He, XM; Radulescu, VD, Small linear perturbations of fractional Choquard equations with critical exponent, J. Differ. Equ., 282, 481-540 (2021) · Zbl 1464.35082
[16] Li, XF; Ma, SW, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 22, 1950023 (2020) · Zbl 1440.35139
[17] Li, GD; Li, YY; Tang, CL, Existence and asymptotic behavior of ground state solutions for Schrödinger equations with Hardy potential and Berestycki-Lions type conditions, J. Differ. Equ., 275, 77-115 (2021) · Zbl 1455.35066
[18] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 93-105 (1976/1977) · Zbl 0369.35022
[19] Lions, PL, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072 (1980) · Zbl 0453.47042
[20] Lions, PL, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-97 (1987) · Zbl 0618.35111
[21] Liu, XN; Ma, SW; Xia, JK, Multiple bound states of higher topological type for semi-classical Choquard equations, Proc. R. Soc. Edinb. Sect. A, 151, 329-355 (2021) · Zbl 1459.35178
[22] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467 (2010) · Zbl 1185.35260
[23] Mattner, L., Strict definiteness of integrals via complete monotonicity of derivatives, Trans. Am. Math. Soc., 349, 3321-3342 (1997) · Zbl 0901.26009
[24] Moroz, V.; Van Schaftingen, J., Existence of groundstates for a class of nonlinear Choquard equations, Trans. Am. Math. Soc., 367, 6557-6579 (2015) · Zbl 1325.35052
[25] Moroz, V.; Van Schaftingen, J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 199-235 (2015) · Zbl 1309.35029
[26] Moroz, V.; Van Schaftingen, J., A guide to the Choquard equation, J. Fixed Point Theory Appl., 19, 1, 773-813 (2017) · Zbl 1360.35252
[27] Pekar, S., Untersuchung über die Elektronentheorie der Kristalle (1954), Berlin: Akademie Verlag, Berlin · Zbl 0058.45503
[28] Penrose, R., Quantum computation, entanglement and state reduction, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 356, 1743, 1927-1939 (1998) · Zbl 1152.81659
[29] Qin, DD; Tang, XH, On the planar Choquard equation with indefinite potential and critical exponential growth, J. Differ. Equ., 285, 40-98 (2021) · Zbl 1465.35249
[30] Qin, DD; Radulescu, VD; Tang, XH, Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations, J. Differ. Equ., 275, 652-683 (2021) · Zbl 1456.35187
[31] Qin, DD; Lai, LZ; Yuan, S.; Wu, QF, Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity, J. Math. Anal. Appl., 500, 125143 (2021) · Zbl 1465.35248
[32] Rabinowitz, PH, Minimax Methods in Critical Point Theory with Applications to Differential Equations (1986), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0609.58002
[33] Rabinowitz, PH, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087
[34] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, Analysis of Operators (1978), New York: Academic Press, New York · Zbl 0401.47001
[35] Schechter, M.; Simon, B., Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77, 482-492 (1980) · Zbl 0458.35024
[36] Seok, J., Nonlinear Choquard equations involving a critical local term, Appl. Math. Lett., 63, 77-87 (2017) · Zbl 1458.35180
[37] Szulkin, A.; Weth, T., The Method of Nehari Manifold, Handbook of Nonconvex Analysis and Applications, 597-632 (2010), Somerville: International Press, Somerville · Zbl 1218.58010
[38] Tang, XH; Lin, XY; Yu, JS, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 31, 369-383 (2019) · Zbl 1414.35062
[39] Tang, XH; Chen, ST; Lin, X.; Yu, JS, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differ. Equ., 268, 4663-4690 (2020) · Zbl 1437.35224
[40] Terracini, S., On positive solutions to a class equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 2, 241-264 (1996) · Zbl 0847.35045
[41] Willem, M., Minimax Theorems (1996), Boston: Birkhäuser, Boston · Zbl 0856.49001
[42] Wu, QF; Qin, DD; Chen, J., Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials, Nonlinear Anal., 197, 111863 (2020) · Zbl 1440.35144
[43] Xia, JK; Wang, Z-Q, Saddle solutions for the Choquard equation, Calc. Var. Partial Differ. Equ., 58, 3, 85 (2019) · Zbl 1418.35161
[44] Xia, JK; Zhang, X., Saddle solutions for the critical Choquard equation, Calc. Var. Partial Differ. Equ., 60, 53 (2021) · Zbl 1459.35216
[45] Zhang, J.; Wu, QF; Qin, DD, Semiclassical solutions for Choquard equations with Berestycki-Lions type conditions, Nonlinear Anal., 188, 22-49 (2019) · Zbl 1429.35093
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