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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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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