Bulletin of the Korean Mathematical Society (대한수학회보)

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GROUND STATE SOLUTIONS OF NON-RESONANT COOPERATIVE ELLIPTIC SYSTEMS WITH SUPERLINEAR TERMS

  • Chen, Guanwei (School of Mathematics and Statistics Anyang Normal University)
  • Received : 2013.04.09
  • Published : 2014.05.31
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Abstract

In this paper, we study the existence of ground state solutions for a class of non-resonant cooperative elliptic systems by a variant weak linking theorem. Here the classical Ambrosetti-Rabinowitz superquadratic condition is replaced by a general super quadratic condition.

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References

  1. G. Chen and S. Ma, Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 271-286. https://doi.org/10.1007/s00526-012-0581-5
  2. G. Chen and S. Ma, Asymptotically or super linear cooperative elliptic systems in the whole space, Sci. China Math. 56 (2013), no. 6, 1181-1194. https://doi.org/10.1007/s11425-013-4567-3
  3. D. G. Costa and C. A. Magalhaes, A variational approach to subquadratic perturbations of elliptic systems, J. Differential Equations 111 (1994), no. 1, 103-122. https://doi.org/10.1006/jdeq.1994.1077
  4. D. G. Costa and C. A. Magalhaes, A unified approach to a class of strongly indefinite functionals, J. Differential Equations 125 (1996), no. 2, 521-547. https://doi.org/10.1006/jdeq.1996.0039
  5. G. Fei, Multiple solutions of some nonlinear strongly resonant elliptic equations without the (PS) condition, J. Math. Anal. Appl. 193 (1995), no. 2, 659-670. https://doi.org/10.1006/jmaa.1995.1259
  6. M. Lazzo, Nonlinear differential problems and Morse thoery, Nonlinear Anal. 30 (1997), no. 1, 169-176. https://doi.org/10.1016/S0362-546X(96)00220-9
  7. S. Li and J. Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math. 25 (1999), no. 3, 563-582.
  8. S. Li and W. Zou, The Computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl. 235 (1999), no. 1, 237-259. https://doi.org/10.1006/jmaa.1999.6396
  9. S. Ma, Infinitely many solutions for cooperative elliptic systems with odd nonlinearity, Nonlinear Anal. 71 (2009), no. 5-6, 1445-1461. https://doi.org/10.1016/j.na.2008.12.012
  10. S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal. 73 (2010), no. 12, 3856-3872. https://doi.org/10.1016/j.na.2010.08.013
  11. A. Pomponio, Asymptotically linear cooperative elliptic system: existence and multiplicity, Nonlinear Anal. 52 (2003), no. 3, 989-1003. https://doi.org/10.1016/S0362-546X(02)00148-7
  12. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
  13. M. Ramos and H. Tavares, Solutions with multiple spike patterns for an elliptic system, Calc. Var. Partial Differential Equations 31 (2008), no. 1, 1-25.
  14. M. Ramos and J. F. Yang, Spike-layered solutions for an elliptic system with Neumann boundary conditions, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3265-3284. https://doi.org/10.1090/S0002-9947-04-03659-1
  15. M. Schechter and W. Zou, Weak linking theorems and Schrodinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var. 9 (2003), 601-619. https://doi.org/10.1051/cocv:2003029
  16. A. Szulkin and T.Weth, Ground state solutions for some indefinite variational problems, J. Functional Analysis 257 (2009), no. 12, 3802-3822. https://doi.org/10.1016/j.jfa.2009.09.013
  17. C.-L. Tang and Q.-J. Gao, Elliptic resonant problems at higher eigenvalues with an unbounded nonlinear term, J. Differential Equations 146 (1998), no. 1, 56-66. https://doi.org/10.1006/jdeq.1998.3411
  18. M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.
  19. W. Zou, Solutions for resonant elliptic systems with nonodd or odd nonlinearities, J. Math. Anal. Appl. 223 (1998), no. 2, 397-417. https://doi.org/10.1006/jmaa.1998.5938
  20. W. Zou, Multiple solutions for asymptotically linear elliptic systems, J. Math. Anal. Appl. 255 (2001), no. 1, 213-229. https://doi.org/10.1006/jmaa.2000.7236
  21. W. Zou and S. Li, Nontrivial Solutions for resonant cooperative elliptic systems via computations of critical groups, Nonlinear Anal. 38 (1999), no. 2, Ser. A: Theory Methods, 229-247. https://doi.org/10.1016/S0362-546X(98)00191-6
  22. W. Zou and S. Li, Infinitely many solutions for Hamiltonian systems, J. Differential Equations 186 (2002), no. 1, 141-164. https://doi.org/10.1016/S0022-0396(02)00005-0

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