Abstract
This paper concerns an elliptic system with critical exponents:
where \(N\ge 3, r\ge 2, 2^*=\frac{2N}{N-2}, \lambda _j\in (0, \frac{(N-2)^2}{4})\) for all \( j=1, \ldots ,r \); \(\beta _{jk}=\beta _{kj}\); \(\alpha _{jk}>1, \alpha _{kj}>1,\) and \(\alpha _{jk}+\alpha _{kj}=2^* \) for all \(k\ne j\). Note that the nonlinearities \(u_j^{2^*-1}\) and the coupling terms are all critical in arbitrary dimension \(N\ge 3 \). The signs of the coupling constants \(\beta _{ij}\) are decisive for the existence of the ground-state solutions. We show that the critical system with \(r\ge 3\) has a positive ground-state solution for all \(\beta _{jk}>0\) with some constraint on \(\lambda _j\). However, there is no ground-state solution when all \(\beta _{jk}\) are negative. It is also proved that the positive solution of the system is radially symmetric. Furthermore, we obtain an uniqueness theorem for the case \(r\ge 3\) with \(N=4\) and an existence theorem for the case \(r=2\) with general coupling exponents.
Similar content being viewed by others
References
Abdellaoui, B., Peral, I., Felli, V.: Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of \({\mathbb{R}}^N\). Adv. Differ. Equ. 9(5–6), 481–508 (2004)
Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)
Ambrosetti, A., Colorado, E., Ruiz, D.: Multi-bump solitons to linearly coupled systems of non- linear Schrödinger equations. Calc. Var. Partial Differ. Equ. 30, 85–112 (2007)
Bartsch, T., Dancer, N., Wang, Z.Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37(3–4), 345–361 (2010)
Bartsch, T., Wang, Z.Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Equ. 19(3), 200–207 (2006)
Bartsch, T., Wang, Z.Q., Wei, J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2(2), 353–367 (2007)
Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)
Byeon, J.: Semi-classical standing waves for nonlinear Schrödinger systems. Calc. Var. Partial Differ. Equ. 54(2), 2287–2340 (2015)
Chang, S., Lin, C.S., Lin, T.C., Lin, W.: Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates. Phys. D 196, 341–361 (2004)
Chen, Z., Lin, C.S., Zou, W.: Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XV, 859–897 (2016)
Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205(2), 515–551 (2012)
Chen, Z., Zou, W.: An optimal constant for the existence of least energy solutions of a coupled Schrödinger system. Calc. Var. Partial Differ. Equ. 48, 695–711 (2013)
Chen, Z., Zou, W.: A remark on doubly critical elliptic systems. Calc. Var. Partial Differ. Equ. 50, 939–965 (2014)
Chen, Z., Zou, W.: Existence and symmetry of positive ground states for a doubly critical Schrödinger system. Trans. Am. Math. Soc. 367(5), 3599–3646 (2015)
Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case. Calc. Var. Partial Differ. Equ. 52(1–2), 423–467 (2015)
Dancer, E.N., Wei, J., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 953–969 (2010)
Felli, V., Pistoia, A.: Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical growth. Commun. Partial Differ. Equ. 31(1–3), 21–56 (2006)
Lin, T.C., Wei, J.: Ground state of \(N\) coupled nonlinear Schrödinger equations in \({{ R}}^n\), \(n\le 3\). Commun. Math. Phys. 255(3), 629–653 (2005)
Lin, T.C., Wei, J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 403–439 (2005)
Lin, T.C., Wei, J.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229(2), 538–569 (2006)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana 1(1), 145–201 (1985)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1(2), 45–121 (1985)
Liu, Z.L., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282, 721–731 (2008)
Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)
Eugenio, P., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. J. Eur. Math. Soc. 10(1), 47–71 (2008)
Noris, B., Tavares, H., Terracini, S., Verzini, G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267–302 (2010)
Peng, S., Wang, Z.: Segregated and synchronized vector solutions for nonlinear Schrödinger systems. Arch. Ration. Mech. Anal. 208(1), 305–339 (2013)
Pomponio, A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 227(1), 258–281 (2006)
Sato, Y., Wang, Z.: Multiple positive solutions for Schrödinger systems with mixed couplings. Calc. Var. Partial Differ. Equ. 54(2), 1373–1392 (2015)
Sato, Y., Wang, Z.: Least energy solutions for nonlinear Schrödinger systems with mixed attractive and repulsive couplings. Adv. Nonlinear Stud. 15, 1–22 (2015)
Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}}^n\). Commun. Math. Phys. 271(1), 199–221 (2007)
Soave, N.: On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition. Calc. Var. Partial Differ. Equ. (2014). https://doi.org/10.1007/s00526-014-0764-3
Soave, N., Tavares, H.: New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms. J. Differ. Equ. 261(1), 505–537 (2016)
Smets, D.: Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities. Trans. Am. Math. Soc. 357(7), 2909–2938 (2005). ((electronic))
Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 4(110), 353–372 (1976)
Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241–264 (1996)
Terracini, S., Verzini, G.: Multipulse phases in \(k\)-mixtures of Bose–Einstein condensates. Arch. Ration. Mech. Anal. 194, 717–741 (2009)
Wang, Z., Willem, M.: Partial symmetry of vector solutions for elliptic systems. J. Anal. Math. 122, 69–85 (2014)
Wei, J., Weth, T.: Nonradial symmetric bound states for a system of coupled Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18(3), 279–293 (2007)
Wei, J., Weth, T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21(2), 305–317 (2008)
Wei, J., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Rat. Mech. Anal. 190, 83–106 (2008)
Wei, J., Yao, W.: Uniqueness of positive solutions to some coupled nonlinear Schrödinger equa- tions. Commun. Pure Appl. Anal. 11, 1003–1011 (2012)
Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston, Inc., Boston (1996)
Acknowledgements
The research of Z. Guo is partially supported by NSFC (no. 11701248) and NFSLN (Research on Klein-Gordon-Maxwell Problem); The research of S. Luo is partially supported by double thousands plan of Jiangxi (no. jxsq2019101048) and NSFC (no. 12001253). The research of W. Zou is partially supported by NSF of China (11801581, 11025106, 11371212, 11271386).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Guo, Z., Luo, S. & Zou, W. On a critical Schrödinger system involving Hardy terms. J. Fixed Point Theory Appl. 23, 53 (2021). https://doi.org/10.1007/s11784-021-00891-z
Accepted:
Published:
DOI: https://doi.org/10.1007/s11784-021-00891-z