## Ground state solutions for the nonlinear Schrödinger-Maxwell equations.(English)Zbl 1147.35091

Summary: We study the nonlinear Schrödinger-Maxwell equations
\begin{aligned} -\Delta u+V(x)u+ \varphi u=|u|^{p-1}u &\quad\text{in }\mathbb R^3,\\ -\Delta\varphi=u^2 &\quad\text{in }\mathbb R^3. \end{aligned}
If $$V$$ is a positive constant, we prove the existence of a ground state solution $$(u,\varphi)$$ for $$2<p<5$$. The non-constant potential case is treated for $$3<p<5$$, and $$V$$ possibly unbounded below. Existence and nonexistence results are proved also when the nonlinearity exhibits a critical growth.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B50 Maximum principles in context of PDEs 49J40 Variational inequalities
Full Text:

### References:

 [1] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., in press [2] Azzollini, A.; Pomponio, A., On a “zero mass” nonlinear Schrödinger equation, Adv. nonlinear stud., 7, 599-627, (2007) · Zbl 1132.35472 [3] Benci, V.; Fortunato, D., An eigenvalue problem for the schrödinger – maxwell equations, Topol. methods nonlinear anal., 11, 283-293, (1998) · Zbl 0926.35125 [4] Benci, V.; Fortunato, D.; Masiello, A.; Pisani, L., Solitons and the electromagnetic field, Math. Z., 232, 73-102, (1999) · Zbl 0930.35168 [5] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations. I. existence of a ground state, Arch. ration. mech. anal., 82, 313-345, (1983) · Zbl 0533.35029 [6] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029 [7] Candela, A.M.; Salvatore, A., Multiple solitary waves for non-homogeneous schrödinger – maxwell equations, Mediterr. J. math., 3, 483-493, (2006) · Zbl 1167.35350 [8] Cassani, D., Existence and non-existence of solitary waves for the critical klein – gordon equation coupled with Maxwell’s equations, Nonlinear anal., 58, 7-8, 733-747, (2004) · Zbl 1057.35041 [9] Cingolani, S.; Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. differential equations, 160, 118-138, (2000) · Zbl 0952.35043 [10] Coclite, G.M., A multiplicity result for the linear schrödinger – maxwell equations with negative potential, Ann. polon. math., 79, 21-30, (2002) · Zbl 1130.35333 [11] Coclite, G.M., A multiplicity result for the nonlinear schrödinger – maxwell equations, Commun. appl. anal., 7, 417-423, (2003) · Zbl 1085.81510 [12] Coclite, G.M.; Georgiev, V., Solitary waves for maxwell – schrödinger equations, Electron. J. differential equations, 94, 1-31, (2004) · Zbl 1064.35180 [13] Coleman, S.; Glaser, V.; Martin, A., Action minima among solutions to a class of Euclidean scalar field equations, Comm. math. phys., 58, 211-221, (1978) [14] 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 [15] D’Aprile, T.; Mugnai, D., Non-existence results for the coupled klein – gordon – maxwell equations, Adv. nonlinear stud., 4, 307-322, (2004) · Zbl 1142.35406 [16] d’Avenia, P., Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. nonlinear stud., 2, 177-192, (2002) · Zbl 1007.35090 [17] Kikuchi, H., On the existence of a solution for elliptic system related to the maxwell – schrödinger equations, Nonlinear anal., 67, 1445-1456, (2007) · Zbl 1119.35085 [18] Lazzo, M., Multiple solutions to some singular nonlinear Schrödinger equations, Electron. J. differential equations, 9, 1-14, (2001) · Zbl 0964.35141 [19] Lions, P.L., The concentration-compactness principle in the calculus of variation. the locally compact case. part I, Ann. inst. H. Poincaré anal. non linéaire, 1, 109-145, (1984) · Zbl 0541.49009 [20] Lions, P.L., The concentration-compactness principle in the calculus of variation. the locally compact case. part II, Ann. inst. H. Poincaré anal. non linéaire, 1, 223-283, (1984) · Zbl 0704.49004 [21] Pisani, L.; Siciliano, G., Neumann condition in the schrödinger – maxwell system, Topol. methods nonlinear anal., 29, 251-264, (2007) · Zbl 1157.35480 [22] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087 [23] Ruiz, D., The schrödinger – poisson equation under the effect of a nonlinear local term, J. funct. anal., 237, 655-674, (2006) · Zbl 1136.35037 [24] Salvatore, A., Multiple solitary waves for a non-homogeneous schrödinger – maxwell system in $$\mathbb{R}^3$$, Adv. nonlinear stud., 6, 157-169, (2006) · Zbl 1229.35065 [25] Talenti, G., Best constant in Sobolev inequality, Ann. mat. pura appl., 110, 353-372, (1976) · Zbl 0353.46018 [26] Wang, Z.; Zhou, H., Positive solution for a nonlinear stationary schrödinger – poisson system in $$\mathbb{R}^3$$, Discrete contin. dyn. syst., 18, 809-816, (2007) · Zbl 1133.35427 [27] Wang, X.F.; Zeng, B., On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. math. anal., 28, 633-655, (1997) · Zbl 0879.35053 [28] Willem, M., Minimax theorems, Progr. nonlinear differential equations appl., vol. 24, (1996), Birkhäuser Boston Boston, MA · Zbl 0856.49001
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.