## Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent.(English)Zbl 1350.35086

Summary: Existence of one positive solution of the generalized Schrödinger-Poisson system $\begin{cases} - \Delta u + V(x) u - K(x) \phi | u |^3 u = f(x, u) \quad & \text{in } \mathbb{R}^3, \\ - \Delta \phi = K(x) | u |^5 \quad & \text{in } \mathbb{R}^3, \end{cases}$ where $$V, K, f$$ are asymptotically periodic functions of $$x$$, is proved by the mountain pass theorem and the concentration-compactness principle. The system with subcritical nonlocal term has been studied extensively in the last twenty years, while the system with critical nonlocal term has seldom been studied. It turns out that new techniques are needed in dealing with the case of critical nonlocal term.

### MSC:

 35J47 Second-order elliptic systems 35B09 Positive solutions to PDEs 35B33 Critical exponents in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:

### References:

 [1] Benci, V.; Fortunato, D., An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11, 283-293, (1998) · Zbl 0926.35125 [2] Lions, P. L., Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-97, (1987) · Zbl 0618.35111 [3] Markowich, P. A.; Ringhofer, C. A.; Schmeiser, C., Semiconductor equations, (1990), Spriner-Verlag Vienna · Zbl 0765.35001 [4] Ambrosetti, A.; Ruiz, D., Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10, 391-404, (2008) · Zbl 1188.35171 [5] Coclite, G. M., A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7, 417-423, (2003) · Zbl 1085.81510 [6] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134, 893-906, (2004) · Zbl 1064.35182 [7] Ianni, I., Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41, 365-385, (2013) · Zbl 1330.35128 [8] Kim, S.; Seok, J., On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14, 1250041, (2012) · Zbl 1263.35197 [9] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674, (2006) · Zbl 1136.35037 [10] Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 90-108, (2008) · Zbl 1147.35091 [11] Jiang, Y. S.; Zhou, H. S., Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251, 582-608, (2011) · Zbl 1233.35086 [12] Wang, Z. P.; Zhou, H. S., Positive solution for a nonlinear stationary Schrödinger-Poisson system in $$\mathbb{R}^3$$, Discrete Contin. Dyn. Syst., 18, 809-816, (2007) · Zbl 1133.35427 [13] Zhao, L. G.; Zhao, F. K., On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346, 155-169, (2008) · Zbl 1159.35017 [14] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 277-320, (2006) · Zbl 1126.35057 [15] Alves, C. O.; Souto, M. A.S.; Soares, S. H.M., Schrödinger-Poisson equations without ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377, 584-592, (2011) · Zbl 1211.35249 [16] Zhang, H.; Xu, J. X.; Zhang, F. B., Positive ground states for asymptotically periodic Schrödinger-Poisson systems, Math. Methods Appl. Sci., 36, 427-439, (2013) · Zbl 1271.35022 [17] Azzollini, A.; d’Avenia, P.; Pomponio, A., On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 779-791, (2010) · Zbl 1187.35231 [18] Cerami, G.; Vaira, G., Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248, 521-543, (2010) · Zbl 1183.35109 [19] D’Aprile, T.; Wei, J. C., On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37, 321-342, (2005) · Zbl 1096.35017 [20] He, X. M.; Zou, W. M., Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53, (2012) · Zbl 1274.81078 [21] Ianni, I., Solutions of the Schrödinger-Poisson problem concentrating on spheres, II, existence, Math. Models Methods Appl. Sci., 19, 877-910, (2009) · Zbl 1187.35236 [22] Ianni, I.; Vaira, G., On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8, 573-595, (2008) · Zbl 1216.35138 [23] Ianni, I.; Vaira, G., Solutions of the Schrödinger-Poisson problem concentrating on spheres, I, necessary conditions, Math. Models Methods Appl. Sci., 19, 707-720, (2009) · Zbl 1173.35687 [24] Li, G. B.; Peng, S. J.; Wang, C. H., Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52, (2011) [25] Liu, Z. L.; Wang, Z.-Q.; Zhang, J. J., Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., (2015), published online [26] Ruiz, D., Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci., 15, 141-164, (2005) · Zbl 1074.81023 [27] Sun, J. T.; Chen, H. B.; Nieto, J. J., On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252, 3365-3380, (2012) · Zbl 1241.35057 [28] Wang, Z. P.; Zhou, H. S., Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $$\mathbb{R}^3$$, Calc. Var. Partial Differential Equations, 52, 927-943, (2015) · Zbl 1311.35300 [29] Yang, M. B.; Zhao, F. K.; Ding, Y. H., On the existence of solutions for Schrödinger-Maxwell systems in $$\mathbb{R}^3$$, Rocky Mountain J. Math., 42, 1655-1674, (2012) · Zbl 1253.35166 [30] Zhao, L. G.; Liu, H. D.; Zhao, F. K., Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255, 1-23, (2013) · Zbl 1286.35103 [31] Azzollini, A.; d’Avenia, P., On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387, 433-438, (2012) · Zbl 1229.35060 [32] Li, F. Y.; Li, Y. H.; Shi, J. P., Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16, 1450036, (2014) · Zbl 1309.35025 [33] Lieb, E. H., Existence and uniqueness of the minimizing solution of choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105, (1976/77) · Zbl 0369.35022 [34] Lins, H. F.; Silva, E. A.B., Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71, 2890-2905, (2009) · Zbl 1167.35338 [35] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063 [36] 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 [37] Willem, M., Minimax theorems, (1996), Birkhauser Boston · Zbl 0856.49001 [38] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36, 437-477, (1983) · Zbl 0541.35029 [39] Jabri, Y., (The Mountain Pass Theorem: Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and its Applications, (2003), Cambridge University Press) · Zbl 1036.49001
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.