Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity. (English) Zbl 1479.35386
The authors study a class of pseudo-relativistic Hartree equations with a Berestycki-Lions-type general hypotheses on the nonlinearity. The existence of a family of localized positive solutions is proved by penalization methods. Next the concentrate of the solutions at the local minimum points of the indefinite potential appearing in the equation is shown.
Reviewer: Leszek Gasiński (Kraków)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
pseudo-relativistic Hartree equation; existence of positive solutions; concentration; penalization methodsReferences:
| [1] | Alves, C. O.; Cassani, D.; Tarsi, C.; Yang, M., Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb{R}^2\), J. Differ. Equ., 261, 1933-1972 (2016) · Zbl 1347.35096 |
| [2] | Alves, C. O.; Nóbrega, A. B.; Yang, M., Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equ., 55 (2016), 28 pp. · Zbl 1347.35097 |
| [3] | Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-346 (1983) · Zbl 0533.35029 |
| [4] | Brezis, H.; Kato, T., Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl., 58, 137-151 (1979) · Zbl 0408.35025 |
| [5] | Byeon, J.; Jeanjean, L., Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185, 185-200 (2007) · Zbl 1132.35078 |
| [6] | Byeon, J.; Jeanjean, L., Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dyn. Syst., 19, 255-269 (2007) · Zbl 1155.35089 |
| [7] | Byeon, J.; Wang, Z. Q., Standing waves with critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differ. Equ., 18, 207-219 (2003) · Zbl 1073.35199 |
| [8] | Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260 (2007) · Zbl 1143.26002 |
| [9] | Cassani, D.; Zhang, J., Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth, Adv. Nonlinear Anal., 8, 1184-1212 (2019) · Zbl 1418.35168 |
| [10] | Chandrasekhar, S., The maximum mass of ideal white dwarfs, Astrophys. J., 74, 81-82 (1931) · Zbl 0002.23502 |
| [11] | Cho, Y.; Ozawa, T., On the semi-relativistic Hartree-type equation, SIAM J. Math. Anal., 38, 1060-1074 (2006) · Zbl 1122.35119 |
| [12] | Cingolani, S.; Secchi, S., Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differ. Equ., 258, 4156-4179 (2015) · Zbl 1319.35204 |
| [13] | Cingolani, S.; Tanaka, K., Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35, 1885-1924 (2019) · Zbl 1431.35169 |
| [14] | Coti Zelati, V.; Nolasco, M., Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam., 29, 1421-1436 (2013) · Zbl 1283.35121 |
| [15] | Coti Zelati, V.; Nolasco, M., Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations, Rend. Lincei Mat. Appl., 22, 51-72 (2011) · Zbl 1219.35292 |
| [16] | Coti Zelati, V.; Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Am. Math. Soc., 4, 693-727 (1991) · Zbl 0744.34045 |
| [17] | D’Avenia, P.; Siciliano, G.; Squassina, M., On fractional Choquard equations, Math. Models Methods Appl. Sci., 25, 1447-1476 (2015) · Zbl 1323.35205 |
| [18] | Ding, Y.; Gao, F.; Yang, M., Semiclassical states for Choquard type equations with critical growth: critical frequency case, Nonlinearity, 33, 6695-6728 (2020) · Zbl 1454.35085 |
| [19] | Du, L.; Yang, M., Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39, 5847-5866 (2019) · Zbl 1425.35025 |
| [20] | Fröhlich, J.; Lenzmann, E., Blow up for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., 60, 1691-1705 (2007) · Zbl 1135.35011 |
| [21] | Fröhlich, J.; Lars, B.; Jonsson, G.; Lenzmann, E., Boson stars as solitary waves, Commun. Math. Phys., 274, 1-30 (2007) · Zbl 1126.35064 |
| [22] | Fröhlich, J.; Lenzmann, E., Blow up for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., 60, 1691-1705 (2007) · Zbl 1135.35011 |
| [23] | Fröhlich, J.; Lenzmann, E., Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Sémin. Équ. Dériv. Partielles, 60, 1-26 (2004) · Zbl 1292.81040 |
| [24] | Gao, F.; da Silva, E.; Yang, M.; Zhou, J., Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. R. Soc. Edinb. A, 150, 921-954 (2020) · Zbl 1437.35213 |
| [25] | Herr, S.; Lenzmann, E., The Boson star equation with initial data of low regularity, Nonlinear Anal., 97, 125-137 (2014) · Zbl 1292.35280 |
| [26] | Lenzmann, E., Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10, 43-64 (2007) · Zbl 1171.35474 |
| [27] | Lenzmann, E., Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. PDE, 2, 1-27 (2009) · Zbl 1183.35266 |
| [28] | Lewin, M.; Lenzmann, E., On singularity formation for the \(L^2\)-critical boson star equation, Nonlinearity, 24, 3515-3540 (2011) · Zbl 1235.35231 |
| [29] | Lieb, E. H., The stability of matter: from atoms to stars, Bull. Am. Math., 22, 1-49 (1990) · Zbl 0698.35135 |
| [30] | Lieb, E. H.; Loss, M., Analysis, Graduate Studies in Mathematics (2001), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0966.26002 |
| [31] | Lieb, E. H.; Thirring, W. E., Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Phys., 115, 494-512 (1984) |
| [32] | Lieb, E. H.; Yau, H. T., The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112, 147-174 (1987) · Zbl 0641.35065 |
| [33] | Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case I. II, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1, 109-145 (1984), 223-283 · Zbl 0541.49009 |
| [34] | Moroz, V.; van Schaftingen, J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 199-235 (2015) · Zbl 1309.35029 |
| [35] | 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 |
| [36] | Mugnai, D., Pseudorelativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13, 799-823 (2013) · Zbl 1298.35199 |
| [37] | Shen, Z.; Gao, F.; Yang, M., Groundstates for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 14, 4082-4098 (2016) · Zbl 1344.35168 |
| [38] | Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces (2007), Springer: Springer Berlin · Zbl 1126.46001 |
| [39] | Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0856.49001 |
| [40] | Yang, M.; Zhang, J.; Zhang, Y., Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Commun. Pure Appl. Anal., 16, 493-512 (2017) · Zbl 1364.35027 |
| [41] | Zheng, Y.; Yang, M.; Shen, Z., On critical pseudo-relativistic Hartree equation with potential well, Topol. Methods Nonlinear Anal., 55, 1, 185-226 (2020) · Zbl 1437.35325 |
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