# 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) $, we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. Citation: Guofeng Che, Haibo Chen, Tsung-fang Wu. Bound state positive solutions for a class of elliptic system with Hartree nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3697-3722. doi: 10.3934/cpaa.2020163 ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation:$-\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. Google Scholar [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France, 49 (1992), 1-139. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] G. Che, H. Shi and Z. Wang, Existence and concentration of positive ground states for a 1-Laplacian problem in$\mathbb{R}^{N}$, Appl. Math. Lett., 100 (2020), Art. 106045. doi: 10.1016/j.aml.2019.106045. Google Scholar [13] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006. Google Scholar [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [15] H. L. Elliott and M. Loss, Analysis, 2$^{nd}$edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [16] G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (2000), 183-240. doi: 10.1137/S0036139997322407. Google Scholar [17] M. Kwong, Uniqueness of positive solutions of$\Delta u-u+u^{p} = 0 $in$\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1997), 93-105. doi: 10.1002/sapm197757293. Google Scholar [19] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar [20] T. C. Lin and J. Wei, Spikes in two–component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011. Google Scholar [21] T.C. Lin and T.F. Wu, Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 33 (2013), 2911-2938. doi: 10.3934/dcds.2013.33.2911. Google Scholar [22] P. L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 1, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109-145. Google Scholar [23] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 223-283. Google Scholar [24] C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2–nodal solutions for semilinear elliptic problems in$\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042. Google Scholar [25] D. F. Lü, A note on Kirchhoff–type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022. Google Scholar [26] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled Schrödinger system, J. Differ. Equ., 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. Google Scholar [27] R. Penrose, On gravity's role in quantum state reduction, Gen. Relat. Gravit., 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar [28] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. Google Scholar [29] Y. Su, New result for nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 102 (2020), Art. 106092. doi: 10.1016/j.aml.2019.106092. Google Scholar [30] J. Sun, H. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. Google Scholar [31] J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002. Google Scholar [32] J. Sun, T. F. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson problems in$\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933. doi: 10.3934/dcds.2018077. Google Scholar [33] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169. doi: 10.1016/j.jmaa.2008.04.053. Google Scholar show all references ##### References:  [1] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation:$-\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. Google Scholar [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France, 49 (1992), 1-139. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] G. Che, H. Shi and Z. Wang, Existence and concentration of positive ground states for a 1-Laplacian problem in$\mathbb{R}^{N}$, Appl. Math. Lett., 100 (2020), Art. 106045. doi: 10.1016/j.aml.2019.106045. Google Scholar [13] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006. Google Scholar [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [15] H. L. Elliott and M. Loss, Analysis, 2$^{nd}$edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [16] G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (2000), 183-240. doi: 10.1137/S0036139997322407. Google Scholar [17] M. Kwong, Uniqueness of positive solutions of$\Delta u-u+u^{p} = 0 $in$\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1997), 93-105. doi: 10.1002/sapm197757293. Google Scholar [19] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. Google Scholar [20] T. C. Lin and J. Wei, Spikes in two–component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011. Google Scholar [21] T.C. Lin and T.F. Wu, Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 33 (2013), 2911-2938. doi: 10.3934/dcds.2013.33.2911. Google Scholar [22] P. L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 1, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109-145. Google Scholar [23] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 223-283. Google Scholar [24] C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2–nodal solutions for semilinear elliptic problems in$\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042. Google Scholar [25] D. F. Lü, A note on Kirchhoff–type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022. Google Scholar [26] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled Schrödinger system, J. Differ. Equ., 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. Google Scholar [27] R. Penrose, On gravity's role in quantum state reduction, Gen. Relat. Gravit., 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar [28] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. Google Scholar [29] Y. Su, New result for nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 102 (2020), Art. 106092. doi: 10.1016/j.aml.2019.106092. Google Scholar [30] J. Sun, H. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. Google Scholar [31] J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002. Google Scholar [32] J. Sun, T. F. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson problems in$\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933. doi: 10.3934/dcds.2018077. Google Scholar [33] L. Zhao and F. 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