## Topological Methods in Nonlinear Analysis

### Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent

#### Abstract

In this paper, we study the existence and nonexistence of positive least energy solutions of the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: \begin{equation*} \begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\left(\frac{1}{|x|^{4}}\ast u^{2}\right)u +\beta \left(\frac{1}{|x|^{4}}\ast v^{2}\right) u, & x \in \Omega,\\ -\Delta v+\nu_{2}v=\mu_{2}\left(\frac{1}{|x|^{4}}\ast v^{2}\right)v +\beta\left(\frac{1}{|x|^{4}}\ast u^{2}\right)v, & x \in \Omega,\\ u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $(-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 623-657.

Dates
First available in Project Euclid: 11 May 2019

https://projecteuclid.org/euclid.tmna/1557540142

Digital Object Identifier
doi:10.12775/TMNA.2019.014

#### Citation

You, Song; Wang, Qingxun; Zhao, Peihao. Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 623--657. doi:10.12775/TMNA.2019.014. https://projecteuclid.org/euclid.tmna/1557540142

#### References

• C.O. Alves, F.S. Gao, M. Squassina and M.B. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations 263 (2017), 3943–3988.
• A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schröodinger equation, C.R. Math. Acad. Sci. Paris 342 (2006), 453–458.
• A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. 75 (2007), 67–82.
• A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007.
• T. Bartsch, N. Dancer and Z.Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345–361.
• T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations 19 (2006), 200–207.
• Z.J. Chen, C.S. Lin and W.M. Zou, Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (2016), 859–897.
• Z.J. Chen and W.M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012), 515–551.
• Z.J. Chen and W.M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations 48 (2013), 695–711.
• S. Correia, F. Oliveira and H. Tavares, Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with $d\geq 3$ equations, J. Funct. Anal. 271 (2016), 2247–2273.
• E.N. Dancer, J.C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 953–969.
• F.S. Gao and M.B. Yang, On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., DOI: 10.1007/s11425-016-9067-5.
• F.S. Gao and M.B. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy–Littlewood–Sobolev inequality, Commun. Contemp. Math., https://doi.org/ 10.1142/S0219199717500377.
• F.S. Gao and M.B. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), 1006–1041.
• C. Le Bris and P.L. Lions, From atoms to crystals: a mathematical journey, Bull. Amer. Math. Soc. 42 (2005), 291–363.
• M. Lewin and J. Sabin, The Hartree equation for infinitely many particles I. Well-posedness theory, Comm. Math. Phys. 334 (2015), 117–170.
• E.H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), 185–194.
• E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics, vol. 14, Amer. Math. Soc., Providence, RI, 2001.
• T.C. Lin and J.C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^{N}$, $n\leq 3$, Comm. Math. Phys. 255 (2005), 629–653.
• T.C. Lin and J.C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403–439.
• P.L. Lions, Solutions of Hartree–Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), 33–97.
• Z.L. Liu and Z.Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys. 282 (2008), 721–731.
• Z.L. Liu and Z.Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud. 10 (2010), 175–193.
• M. Mitchell, Z.G. Chen, M.F. Shih and M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett. 77 (1996), 490–493.
• M. Mitchell and M. Segev, Self-trapping of partially spatially incoherent light, Nature 387 (1997), 880–883.
• V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 5 (2015), 1550005.
• V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184.
• B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math. 63 (2010), 267–302.
• S. Peng and Z.Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal. 208 (2013), 305–339.
• B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{n}$, Comm. Math. Phys. 271 (2007), 199–221.
• M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, fourth edition, Springer–Verlag, Berlin, 2008.
• J. Van Schaftingen and J.K. Xia, Standing waves with a critical frequency for nonlinear Choquard equations, Nonlinear Anal. 161 (2017), 87–107.
• J. Wang, Y.Y. Dong, Q. He and L. Xiao, Multiple positive solutions for a coupled nonlinear Hartree type equations with perturbations, J. Math. Anal. Appl. 450 (2017),780–794.
• J. Wang and J.P. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, DOI: 10.1007/s00526-017-1268-8.
• J.C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 (2008), 83–106.
• M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, MA, 1996.
• M.B. Yang, Y.H. Wei and Y.H. Ding, Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities, Z. Angew. Math. Phys. 65 (2014), 41–68.
• S. You, P.H. Zhao and Q.X. Wang, Positive least energy solutions for coupled nonlinear Choquard equations in $\mathbb{R}^{N}$. (to appear)