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

Song You, Qingxun Wang, Peihao Zhao

DOI: http://dx.doi.org/10.12775/TMNA.2019.014

Abstract


\begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\bigg(\frac{1}{|x|^{4}}\ast u^{2}\bigg)u +\beta \bigg(\frac{1}{|x|^{4}}\ast v^{2}\bigg)u, & x \in \Omega,\\[10pt] \displaystyle -\Delta v+\nu_{2}v=\mu_{2}\bigg(\frac{1}{|x|^{4}}\ast v^{2}\bigg)v +\beta\bigg (\frac{1}{|x|^{4}}\ast u^{2}\bigg)v, & x \in \Omega,\\[10pt] 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.

Keywords


Coupled Choquard equations; least energy solutions; Hardy--Littlewood--Sobolev critical exponent

Full Text:

PREVIEW FULL TEXT

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 ProbCambridge, 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 ≥ 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 RN , n ≤ 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 Rn , 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/s00526-0171268-8.

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 RN . (to appear)


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism