# American Institute of Mathematical Sciences

July  2020, 19(7): 3697-3722. doi: 10.3934/cpaa.2020163

## Bound state positive solutions for a class of elliptic system with Hartree nonlinearity

 1 School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, Guangdong, China 2 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China 3 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

*Corresponding author

Received  August 2019 Revised  January 2020 Published  April 2020

Fund Project: H. Chen was supported by the National Natural Science Foundation of China (Grant No. 11671403). T. F. Wu was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 108-2115-M-390-007-MY2)

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:
 $\begin{equation*} \begin{cases} -\varepsilon ^{2}\Delta u+V_{1}\left( x\right) u+\lambda _{1}\left(I_{\varepsilon }\ast |u|^{p+1}\right)|u|^{p-1}u\\ \qquad\quad\; = \left(\mu _{1}|u|^{2p}+\beta(x) |u|^{q-1}|v|^{q+1}\right)u, & \text{in }\mathbb{R}^{N}, \\ -\varepsilon ^{2}\Delta v+V_{2}\left( x\right) v+\lambda _{2}\left(I_{\varepsilon }\ast |v|^{p+1}\right)|v|^{p-1}v\\ \qquad\quad\; = \left(\mu _{2}|v|^{2p}+\beta(x) |v|^{q-1}|u|^{q+1}\right)v, & \text{in }\mathbb{R}^{N}\,, \\ u,v\in H^{1}(\mathbb{R}^{N}),\quad u,v>0, \end{cases} \end{equation*}$
where
 $0<\varepsilon \ll 1$
is a small parameter,
 $0 , $ I_{\varepsilon}(x) = \frac{\Gamma((N-\alpha)/2)} {\Gamma(\alpha/2)\pi^{\frac{N}{2}}2^{\alpha}\varepsilon^{\alpha}}\frac{1}{|x|^{N-\alpha}}, \; x\in\mathbb{R}^{N}\setminus\{0\} $, $ \alpha\in(0,N),\; N = 3,4,5 $and $ \lambda _{l}\geq0,\; \mu _{l}>0,\; l = 1,2, $are constants. Under some suitable assumptions on the potentials $ V_{l}(x),l = 1,2, $and the coupled function $ \beta(x) $, we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. Citation: Guofeng Che, Haibo Chen, Tsung-fang Wu. Bound state positive solutions for a class of elliptic system with Hartree nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3697-3722. doi: 10.3934/cpaa.2020163 ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation:$-\triangle u+u = a(x)u^{p}+f(x)$in$\mathbb{R}^{N}$, Calc. Var. Partial Differ. Equ., 11 (2000), 63-95. doi: 10.1007/s005260050003. [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France, 49 (1992), 1-139. [3] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [4] H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [5] H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113. [6] F. E. Browder, Lusternik-Schnirelman category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648. doi: 10.1090/S0002-9904-1965-11378-7. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [8] G. Che and H. Chen, Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity, Appl. Math. Lett., 81 (2018), 21-26. doi: 10.1016/j.aml.2017.12.014. [9] G. Che and H. Chen, Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials, Topol. Meth. Nonlinear Anal., 51 (2018), 215-242. doi: 10.12775/tmna.2017.046. [10] G. Che, H. Chen and T. F. Wu, Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling, J. Math. Phys, 60 (2019), Art. 081511. doi: 10.1063/1.5087755. [11] G. Che, H. Chen and L. Yang, Existence and multiplicity of solutions for semilinear elliptic systems with periodic potential, Bull. Malays. Math. Sci. Soc., 42 (2019), 1329-1348. doi: 10.1007/s40840-017-0551-3. [12] G. Che, H. Shi and Z. Wang, Existence and concentration of positive ground states for a 1-Laplacian problem in$\mathbb{R}^{N}$, Appl. Math. Lett., 100 (2020), Art. 106045. doi: 10.1016/j.aml.2019.106045. [13] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006. [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [15] H. L. Elliott and M. Loss, Analysis, 2$^{nd}$edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [16] G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (2000), 183-240. doi: 10.1137/S0036139997322407. [17] M. Kwong, Uniqueness of positive solutions of$\Delta u-u+u^{p} = 0 $in$\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1997), 93-105. doi: 10.1002/sapm197757293. [19] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [20] T. C. Lin and J. Wei, Spikes in two–component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011. [21] T.C. Lin and T.F. Wu, Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 33 (2013), 2911-2938. doi: 10.3934/dcds.2013.33.2911. [22] P. L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 1, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109-145. [23] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 223-283. [24] C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2–nodal solutions for semilinear elliptic problems in$\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042. [25] D. F. Lü, A note on Kirchhoff–type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022. [26] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled Schrödinger system, J. Differ. Equ., 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. [27] R. Penrose, On gravity's role in quantum state reduction, Gen. Relat. Gravit., 28 (1996), 581-600. doi: 10.1007/BF02105068. [28] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [29] Y. Su, New result for nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 102 (2020), Art. 106092. doi: 10.1016/j.aml.2019.106092. [30] J. Sun, H. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. [31] J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002. [32] J. Sun, T. F. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson problems in$\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933. doi: 10.3934/dcds.2018077. [33] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169. doi: 10.1016/j.jmaa.2008.04.053. show all references ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation:$-\triangle u+u = a(x)u^{p}+f(x)$in$\mathbb{R}^{N}$, Calc. Var. Partial Differ. Equ., 11 (2000), 63-95. doi: 10.1007/s005260050003. [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France, 49 (1992), 1-139. [3] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [4] H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [5] H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113. [6] F. E. Browder, Lusternik-Schnirelman category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648. doi: 10.1090/S0002-9904-1965-11378-7. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [8] G. Che and H. Chen, Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity, Appl. Math. Lett., 81 (2018), 21-26. doi: 10.1016/j.aml.2017.12.014. [9] G. Che and H. Chen, Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials, Topol. Meth. Nonlinear Anal., 51 (2018), 215-242. doi: 10.12775/tmna.2017.046. [10] G. Che, H. Chen and T. F. Wu, Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling, J. Math. Phys, 60 (2019), Art. 081511. doi: 10.1063/1.5087755. [11] G. Che, H. Chen and L. Yang, Existence and multiplicity of solutions for semilinear elliptic systems with periodic potential, Bull. Malays. Math. Sci. Soc., 42 (2019), 1329-1348. doi: 10.1007/s40840-017-0551-3. [12] G. Che, H. Shi and Z. Wang, Existence and concentration of positive ground states for a 1-Laplacian problem in$\mathbb{R}^{N}$, Appl. Math. Lett., 100 (2020), Art. 106045. doi: 10.1016/j.aml.2019.106045. [13] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006. [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [15] H. L. Elliott and M. Loss, Analysis, 2$^{nd}$edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [16] G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (2000), 183-240. doi: 10.1137/S0036139997322407. [17] M. Kwong, Uniqueness of positive solutions of$\Delta u-u+u^{p} = 0 $in$\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1997), 93-105. doi: 10.1002/sapm197757293. [19] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [20] T. C. Lin and J. Wei, Spikes in two–component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011. [21] T.C. Lin and T.F. Wu, Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 33 (2013), 2911-2938. doi: 10.3934/dcds.2013.33.2911. [22] P. L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 1, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109-145. [23] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 223-283. [24] C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2–nodal solutions for semilinear elliptic problems in$\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042. [25] D. F. Lü, A note on Kirchhoff–type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022. [26] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled Schrödinger system, J. Differ. Equ., 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. [27] R. Penrose, On gravity's role in quantum state reduction, Gen. Relat. Gravit., 28 (1996), 581-600. doi: 10.1007/BF02105068. [28] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [29] Y. Su, New result for nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 102 (2020), Art. 106092. doi: 10.1016/j.aml.2019.106092. [30] J. Sun, H. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. [31] J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002. [32] J. Sun, T. F. Wu and Z. 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