Existence and concentration of solutions for singularly perturbed doubly nonlocal elliptic equations. (English) Zbl 1437.35308


35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI


[1] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z.248 (2004) 423-443. · Zbl 1059.35037
[2] Alves, C. O. and Yang, M., Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations257 (2014) 4133-4164. · Zbl 1309.35036
[3] Berestycki, H. and Lions, P. L., Nonlinear scalar field equations, I: Existence of a ground state, Arch. Ration. Mech. Anal.82 (1983) 313-345. · Zbl 0533.35029
[4] Byeon, J. and Jeanjean, L., Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal.185 (2007) 185-200. · Zbl 1132.35078
[5] Byeon, J. and Jeanjean, L., Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dynam. Syst.19 (2007) 255-269. · Zbl 1155.35089
[6] Byeon, J. and Wang, Z.-Q., Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations18 (2003) 207-219. · Zbl 1073.35199
[7] Cingolani, S. and Secchi, S., Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations258 (2015) 4156-4179. · Zbl 1319.35204
[8] Cingolani, S., Secchi, S. and Squassina, M., Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A140 (2010) 973-1009. · Zbl 1215.35146
[9] del Pino, M. and Felmer, P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations4 (1996) 121-137. · Zbl 0844.35032
[10] del Pino, M. and Felmer, P., Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaŕe Anal. Non Linéaire15 (1998) 127-149. · Zbl 0901.35023
[11] Figueiredo, G. M., Ikoma, N. and Júnior, J. R. S., Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal.213 (2014) 931-979. · Zbl 1302.35356
[12] Floer, A. and Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal.69 (1986) 397-408. · Zbl 0613.35076
[13] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 2nd edition, , Vol. 224 (Springer, Berlin, 1983). · Zbl 0562.35001
[14] Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations21 (1996) 787-820. · Zbl 0857.35116
[15] He, Y. and Li, G., Standing waves for a class of Kirchhoff type problems in \(\mathbb{R}^3\) involving critical Sobolev exponents, Calc. Var. Partial Differential Equations54 (2015) 3067-3106. · Zbl 1328.35046
[16] He, Y., Li, G. and Peng, S., Concentrating bound states for Kirchhoff type problem in \(\mathbb{R}^3\) involving critical Sobolev exponents, Adv. Nonlinear Stud.14 (2014) 441-468.
[17] He, X. and Zou, W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\), J. Differential Equations252 (2012) 1813-1834. · Zbl 1235.35093
[18] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal.28 (1997) 1633-1659. · Zbl 0877.35091
[19] Kang, X. and Wei, J., On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations5 (2000) 899-928. · Zbl 1217.35065
[20] Kirchhoff, G., Mechanik (Teubner, Leipzig, 1883).
[21] Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math.57 (1977) 93-105. · Zbl 0369.35022
[22] Lieb, E. H. and Loss, M., Analysis, 2nd edition, , Vol. 14 (American Mathematical Society, 2001). · Zbl 0966.26002
[23] Lieb, E. H. and Simon, B., The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys.53 (1977) 185-194.
[24] Lions, J. L., On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud.30 (1978) 284-346.
[25] Lions, P. L., The Choquard equation and related questions, Nonlinear Anal.4 (1980) 1063-1072. · Zbl 0453.47042
[26] Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire1 (1984) 223-283. · Zbl 0704.49004
[27] Lü, D. and Peng, S., Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems, J. Differential Equations263 (2017) 8947-8978. · Zbl 1376.35051
[28] Ma, L. and Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal.195 (2010) 455-467. · Zbl 1185.35260
[29] Moroz, V. and Van Schaftingen, J., Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal.265 (2013) 153-184. · Zbl 1285.35048
[30] Moroz, V. and Van Schaftingen, J., Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc.367 (2015) 6557-6579. · Zbl 1325.35052
[31] Moroz, V. and Van Schaftingen, J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations52 (2015) 199-235. · Zbl 1309.35029
[32] Oh, Y.-G., Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Comm. Math. Phys.121 (1989) 11-33. · Zbl 0693.35132
[33] Oh, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys.131 (1990) 223-253. · Zbl 0753.35097
[34] Pucci, P. and Serrin, J., A general variational identity, Indiana Univ. Math. J.35 (1986) 681-703. · Zbl 0625.35027
[35] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys.43 (1992) 270-291. · Zbl 0763.35087
[36] Sun, X. and Zhang, Y., Multi-peak solution for nonlinear magnetic Choquard type equation, J. Math. Phys.55 (2014) 031508, 25 pp. · Zbl 1296.35180
[37] Wang, X., On the concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys.153 (1993) 229-244. · Zbl 0795.35118
[38] Wang, J., Tian, L., Xu, J. and Zhang, F., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations253 (2012) 2314-2351. · Zbl 1402.35119
[39] Yang, M., Zhang, J. and Zhang, Y., Multi-peak solution for nonlinear Choquard equation with a general nonlinearity, Commun. Pure Appl. Anal.16 (2017) 493-512. · Zbl 1364.35027
[40] Zelati, V. C. and Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc.4 (1991) 693-727. · Zbl 0744.34045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.