Advertisement

Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential

  • Guofeng Che
  • Haibo ChenEmail author
  • Liu Yang
Article

Abstract

In this paper, we consider the following semilinear elliptic systems:
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$
where \(V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)\) and \(F_{v}(x,u,v)\) are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of \(-\triangle +V\). Under appropriate assumptions on \(F_{u}(x, u, v)\) and \(F_{v}(x, u, v)\), we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.

Keywords

Semilinear elliptic systems Strongly indefinite functional Ground state Nehari–Pankov manifold Variational methods 

Mathematics Subject Classification

35B38 35J20 

References

  1. 1.
    Silva, E.: Existence and multiplicity of solutions for semilinear elliptic systems. NoDEA 1, 339–363 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Duan, S., Wu, X.: The existence of solutions for a class of semilinear elliptic systems. Nonlinear Anal. 73, 2842–2854 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Qu, Z., Tang, C.: Existence and multiplicity results for some elliptic systems at resonance. Nonlinear Anal. 71, 2660–2666 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Shi, H.X., Chen, H.B.: Ground state solutions for resonant cooperative elliptic systems with general superlinear terms. Mediterr. J. Math. 13, 2897–2909 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhang, J., Zhang, Z.: Existence results for some nonlinear elliptic systems. Nonlinear Anal. 71, 2840–2846 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, G., Tang, X.: Nehari-type state solutions for Schrödinger equations including critical exponent. Appl. Math. Lett. 37, 101–106 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Che, G.F., Chen, H.B.: Multiplicity of small negative-energy solutions for a class of semilinear elliptic systems. Bound. Value. Probl. 2016, 1–12 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liao, F.F., Tang, X.H., Qin, D.D.: Super-quadratic conditions for periodic elliptic system on \(\mathbb{R}^{N}\). Electron. J. Differ. Equ. 127, 1–11 (2015)Google Scholar
  9. 9.
    Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743–767 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, J., Tang, X.H., Zhang, W.: Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms. Nonlinear Anal. 95, 1–10 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhang, J., Qin, W.P., Zhao, F.K.: Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. J. Math. Anal. Appl. 399, 433–441 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, G.W., Ma, S.W.: Nonexistence and multiplicity of solutions for nonlinear elliptic systems in \(\mathbb{R}^{N}\). Nonlinear Anal. 36, 233–248 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhang, W., Zhang, J., Zhao, F.K.: Multiple solutions for asymptotically quadratic and superquadratic elliptic system of Hamiltonian type. Appl. Math. Comput. 263, 36–46 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Shi, H.X., Chen, H.B.: Ground state solutions for asymptotically periodic coupled Kirchhoff-type systems with critical growth. Math. Methods Appl. Sci. 39, 2193–2201 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Che, G.F., Chen, H.B.: Existence and multiplicity of systems of Kirchhoff-type equations with general potentials. Math. Methods Appl. Sci. 40, 775–785 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tang, X.H.: Ground state solutions of Nehari–Pankov type for a superlinear Hamiltonian elliptic system on \(\mathbb{R}^{N}\). Canad. Math. Bull. 58(3), 651–663 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Analysis of Operators, vol. IV. Academic Press, New York (1978)zbMATHGoogle Scholar
  18. 18.
    Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pankov, A.: On decay of solutions to nonlinear Schrödinger equations. Proc. Am. Math. Soc. 136, 2565–2570 (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mederski, J.: Ground states of a system of nonlinear Schrödinger equations with periodic potentials. Commun. Partial Diff. Equ. 41(9), 1426–1440 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, Q., Mederski, J.: Ground states of nonlinear Schrödinger equations with sum of periodic and inverse square potentials. J. Differ. Equ. 260, 4180–4202 (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Mederski, J.: Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum. Topol. Methods Nonlinear Anal. 46(2), 755–771 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bieganowski, B., Mederski, J.: Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. arXiv preprint arXiv:1602.05078
  24. 24.
    Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313(1), 15–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pankov, A.: Periodic nonlinear Schröinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Badiale, M., Pisani, L., Rolando, S.: Sum of weighted Lebesgue spaces and nonlinear elliptic equations. Nodea Nonlinear Differ. Equ. Appl. 18, 369–405 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Bartsch, T., Mederski, J.: Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain. Arch. Ration. Mech. Anal. 215(1), 283–306 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Willem, M.: Minimax Theorems. Birkhäuser Verlag, Basel (1996)CrossRefzbMATHGoogle Scholar
  29. 29.
    Struwe, M.: Variational Methods. Springer, Berlin (2008)zbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHengyang Normal UniversityHengyangPeople’s Republic of China

Personalised recommendations