Ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation. (English) Zbl 1436.35118

Summary: This paper deals with the following Choquard equation with a local nonlinear perturbation: \[ \begin{cases} - \Delta u + u = \left( I_\alpha \ast | u |^{\frac{ \alpha}{ N} + 1}\right) | u |^{\frac{ \alpha}{ N} - 1} u + f (u ), \quad x \in \mathbb{R}^N; \\ u \in H^1 (\mathbb{R}^N), \end{cases}\] where \(I_\alpha : \mathbb{R}^N \to \mathbb{R}\) is the Riesz potential, \(N \geq 3\), \(\alpha \in (0, N )\), the exponent \(\frac{ \alpha}{ N} + 1\) is critical with respect to the Hardy-Littlewood-Sobolev inequality, and the nonlinear perturbation \(f\) is only required to satisfy some weak assumptions near 0 and \(\infty\). Our results improve the previous related ones in the literature.


35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear 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, 423-443 (2004) · Zbl 1059.35037
[2] Alves, C. O.; N’obrega, A. B.; Yang, M. B., Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55, 1-28 (2016) · Zbl 1347.35097
[3] Alves, C. O.; Yang, M., Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257, 4133-4164 (2014) · Zbl 1309.35036
[4] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[5] Cassani, D.; Zhang, J., Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth, Adv. Nonlinear Anal., 8, 1184-1212 (2019) · Zbl 1418.35168
[6] Chen, S. T.; Fiscella, A.; Pucci, P.; Tang, X. H., Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268, 2672-2716 (2020) · Zbl 1436.35078
[7] Chen, S. T.; Tang, X. H., Berestycki-lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9, 496-515 (2020) · Zbl 1422.35023
[8] Chen, S. T.; Tang, X. H., Ground state solutions for general Choquard equations with a variable potential and a local nonlinearity, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020) · Zbl 1437.35209
[9] Chen, S. T.; Tang, X. H., On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268, 945-976 (2020) · Zbl 1431.35030
[10] Fiscella, A., A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal., 8, 645-660 (2019) · Zbl 1419.35035
[11] Gao, F.; Yang, M., On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448, 1006-1041 (2017) · Zbl 1357.35106
[12] Goel, D.; Sreenadh, K., Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains, Adv. Nonlinear Anal., 8, 803-835 (2019) · Zbl 1430.35100
[13] Ji, C.; Fang, F.; Zhang, B., A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal., 8, 267-277 (2019) · Zbl 1419.35037
[14] Li, X. F.; Ma, S. W.; Zhang, G., Existence and qualitative properties of solutions for Choquard equations with a local term, 45, 1-25 (2019) · Zbl 1412.35123
[15] Li, G. D.; Tang, C. L., Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 76, 2635-2647 (2018)
[16] Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105 (1976) · Zbl 0369.35022
[17] Lieb, E. H.; Loss, M.; Analysis, G., Graduate Studies in Mathematics, vol. 14 (1997), American Mathematical Society: American Mathematical Society Providence, RI
[18] Lions, P. L., The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072 (1980) · Zbl 0453.47042
[19] Lions, P. L., The concentration-compactness principle in the calculus of variations. the locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 109-145 (1984) · Zbl 0541.49009
[20] Liu, X. N.; Ma, S. W.; Zhang, X., Infinitely many bound state solutions of Choquard equations with potentials, Z. Angew. Math. Phys., 69, 29 (2018) · Zbl 1401.35055
[21] Luo, H., Ground state solutions of Pohozaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467, 842-862 (2018) · Zbl 1398.35071
[22] Mingqi, X.; Rãdulescu, V. D.; Zhang, B., Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities, ESAIM Control Optim. Calc. Var., 24, 1249-1273 (2018) · Zbl 1453.35184
[23] Mingqi, X.; Rãdulescu, V. D.; Zhang, B., A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21, 4, Article 185004 pp. (2019), 36 pages · Zbl 1416.49012
[24] Moroz, I. M.; Penrose, R.; Tod, P., Spherically-symmetric solutions of the Schrödinger-Newton equations, Class. Quantum Gravity, 15, 2733-2742 (1998) · Zbl 0936.83037
[25] Moroz, V.; Van Schaftingen, J., Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184 (2013) · Zbl 1285.35048
[26] Moroz, V.; Van Schaftingen, J., Existence of groundstate for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367, 6557-6579 (2015) · Zbl 1325.35052
[27] Moroz, V.; Van Schaftingen, J., Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17, Article 1550005 pp. (2015), 12 pp · Zbl 1326.35109
[28] Moroz, V.; Van Schaftingen, J., A guide to the Choquard equation, J. Fixed Point Theory Appl., 19, 773-813 (2017) · Zbl 1360.35252
[29] Pekar, S., Untersuchung über die Elektronentheorie der Kristalle (1954), Akademie Verlag: Akademie Verlag Berlin · Zbl 0058.45503
[30] Ruiz, D.; Van Schaftingen, J., Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations, 264, 1231-1262 (2018) · Zbl 1377.35011
[31] Seok, J., Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal., 8, 1083-1098 (2019) · Zbl 1419.35066
[32] Tang, X. H.; Chen, S. T., Ground state solutions of Nehari-Pohoz̆aev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56, 110-134 (2017) · Zbl 1376.35056
[33] Tang, X. H.; Chen, S. T., Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9, 413-437 (2020) · Zbl 1421.35068
[34] Tang, X. H.; Chen, S. T.; Lin, X.; Yu, J. S., Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268, 4663-4690 (2020) · Zbl 1437.35224
[35] Tang, X. H.; Lin, X. Y., Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems, Sci. China Math., 63, 113-134 (2020) · Zbl 1444.35052
[36] Van Schaftingen, J.; Xia, J., Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464, 1184-1202 (2018) · Zbl 1398.35094
[37] Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24 (1996), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA
[38] Xiang, M.; Zhang, B.; Rãdulescu, V. D., Superlinear Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal., 9, 690-709 (2020) · Zbl 1427.35340
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