Topological Methods in Nonlinear Analysis

The existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness

Guanggang Liu, Shaoyun Shi, and Yucheng Wei

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In this paper, we show the existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness, by using the technique of penalized functionals and an infinite dimensional Morse theory developed by Kryszewski and Szulkin. Two applications are given to Hamiltonian systems and elliptic systems.

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Topol. Methods Nonlinear Anal., Volume 43, Number 2 (2014), 323-344.

First available in Project Euclid: 11 April 2016

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Liu, Guanggang; Shi, Shaoyun; Wei, Yucheng. The existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness. Topol. Methods Nonlinear Anal. 43 (2014), no. 2, 323--344.

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  • \refA. Abbondandolo, A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces , Topol. Methods Nonlinear Anal., 9 (1997), 325–382 \ref ––––, Morse theory for asymptotcally linear Hamiltonian systems , Nonlinear Anal., 39 (2000), 997–1049
  • \refP. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorem and applications to nonlinear problems with “strong” resonance at infinity , Nonlinear Anal., 7 (1983), 981–1012
  • \refK.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser, Boston (1993)
  • \refK.C. Chang, J.Q. Liu and M.J. Liu, Nontrivial periodic solutions for strong resonance Hamiltonian systems , Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 103–117 \ref
  • D.G. Costa and C.A. Magalh$\tilde{\text{\plsmc a}}$es, A variational method to subquadratic perturbations of elliptic systems , J. Differential Equations, 111 (1994), 103–122
  • \refD.G. de Figueiredo and P. Felmer, On superqadratic elliptic systems , Trans. Amer. Math. Soc., 111 (1994), 99–116
  • \refY.X. Guo, Morse theury for strongly indefinite functional and its applications, Doctoral thesis, Institute of Mathematics, Peking University (1999) \ref––––, Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance , J. Differential Equations, 175 (2001), 71–87 \ref ––––, Nontrivial solutions for resonant noncooperative elliptic systems , Comm. Pure Appl. Math., 53 (2000), 1335–1349
  • \refJ. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure , J. Func. Anal., 114 (1993), 32–58
  • \ref W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications , Trans. Amer. Math. Soc., 349 (1997), 3181–3234
  • \ref\no13 E. Landesman and A.C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance , J. Math. Mech., 19 (1970), 609–623
  • \ref\no14 Z.L. Liu, J.B. Su and Z.-Q. Wang, Solutions of elliptic problems with nonlinearities of linear growth , Calc. Var. Partial Differential Equations, 35 (2009), 463–480
  • \ref\no15 A. Masiello and L. Pisani, Asymptotically linear elliptic problems at resonance , Ann. Math. Pura Appl., 4 (1996), 1–13
  • \ref J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer–Verlag, New York (1989)
  • \ref\no17 A. Pomponio, An asymptotically linear non-cooperative elliptic system with lack of compactness , Proc. Roy. Soc. Londondon A, 459 (2003), 2265–2279
  • \ref \no18P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, AMS Conf. Ser. Math., 65 (1986)
  • \ref \no19 A. Salvatore, Periodic solutions of asymptotically linear systems without symmetry , Rend. Sem. Math. Univ., 74 (1985), 147–161
  • \ref\no20 M. Solimini, Morse index estimates in minimax theores , Manus. Math., 63 (1989), 421–453
  • \ref\no21 J.B. Su and Z.L. Liu, Bounded resonance problem for elliptic equations , Discrete Contin. Dyn. Syst, 19 (2007), 431–445
  • \ref\no22 A. Szulkin, Cohomology and Morse theory for strong indefinite functionals , Math. Z., 209 (1992), 375–418
  • \ref\no23 A. Szulkin and W.M. Zou, Infinite dimensional cohomology groups and periodic solutions of asymptotically linear Hamiltonian systems , J. Differentail Equations, 174 (2001), 369–391
  • \ref\no24 P.A. Zeng, J.Q. Liu and Y.X. Guo, Computations of critical groups and applications to asymptotically linear wave equation and beam equation , J. Math. Anal. Appl., 300 (2004), 102–128
  • \ref\no25 W. M. Zou, Computations of the cohomology groups with applications to asymptotically linear beam equtions and noncooperative elliptic systems , Comm. Partial Differential Equations, 27 (2002), 115–147