AIMS Mathematics, 2020, 5(4): 3825-3839. doi: 10.3934/math.2020248.

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Multiple solutions to a quasilinear Schrödinger equation with Robin boundary condition

1 Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China
2 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
3 Shanghai University of Medicine and Health Sciences, Shanghai, 201318, China

We study a quasilinear Schrödinger equation with Robin boundary condition. Using the variational methods and the truncation techniques, we prove the existence of two positive solutions when the parameter λ is large enough. We also establish the existence of infinitely many high energy solutions by using Fountain Theorem when λ > 1.
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Keywords quasilinear Schrödinger equation; Robin boundary; multiple solutions; Fountain Theorem

Citation: Yin Deng, Gao Jia, Fanglan Li. Multiple solutions to a quasilinear Schrödinger equation with Robin boundary condition. AIMS Mathematics, 2020, 5(4): 3825-3839. doi: 10.3934/math.2020248


  • 1. X. W. Li, G. Jia, Multiplicity of solutions for quasilinear elliptic problems involving Φ-Laplacian operator and critical growth, Electron. J. Qual. Theory Differ. Equ., 6 (2019), 1-15.
  • 2. N. S. Papageorgiou, V. D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.    
  • 3. N. S. Papageorgiou, V. D. Rădulescu, Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Revista Mat. Complut., 29 (2016), 91-126.
  • 4. N. S. Papageorgiou, V. D. Rădulescu, D. D. Repovš, Robin problems with a general potential and a superlinear reaction, J. Differential Equations, 263 (2017), 3244-3290.    
  • 5. N. S. Papageorgiou, V. D. Rădulescu, D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst., 37 (2017), 2589-2618.    
  • 6. N. S. Papageorgiou, V. D. Rădulescu, Positive solutions of nonlinear Robin eigenvalue problems, Proc. Amer. Math. Soc., 144 (2016), 4913-4928.    
  • 7. M. Poppenberg, K. Schmitt, Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.    
  • 8. J. Q. Liu, Y. Q. Wang, Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.    
  • 9. M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.    
  • 10. S. Liu, J. Zhou, Standing waves for quasilinear Schrödinger equations with indefinite potentials, J. Differential Equations, 265, (2018), 3970-3987.
  • 11. D. Motreanu, V. V. Motreanu, N. S. Papageorgiou, Topological and variational me thods with applications to nonlinear boundary value problems, Springer, New York, 2013.
  • 12. S. Liu, S. J. Li, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.    
  • 13. P. Winkert, L estimates for nonlinear elliptic Neumann boundary value problems, Nonlin. Differ. Equations Appl., 17 (2010), 289-302.    
  • 14. G. M. Liberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1209.    
  • 15. J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.    
  • 16. M. Willem, Minimax Theorems, Birkhäuser: Progress in Nonlinear Differential Equations and Their Applications, 1996.
  • 17. G. D'Aguì, S. Marano, N.S. Papageorgiou, Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl., 433 (2016), 1821-1845.    
  • 18. S. Hu, N. S. Papageorgiou, Positive solutions for Robin problems with general potential and logistic reaction, Comm. Pure. Appl. Anal., 15 (2016), 2489-2507.    
  • 19. S. A. Marano, N. S. Papageorgiou,On a Robin problem with p-Laplacian and reaction bounded only from above, Monatsh. Math., 180 (2016), 317-336.    
  • 20. R. E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998.


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