AIMS Mathematics, 2020, 5(4): 3634-3645. doi: 10.3934/math.2020235.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Infinitely many solutions for a class of biharmonic equations with indefinite potentials

1 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, P. R. China
2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P. R. China

In this paper, we consider the following sublinear biharmonic equations\begin{equation*} \Delta^2 u + V\left( x \right)u =K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*}where $N\geq5,~0<p<1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.
  Figure/Table
  Supplementary
  Article Metrics

Keywords biharmonic equation; indefinite potential; symmetric Mountain Pass Theorem

Citation: Wen Guan, Da-Bin Wang, Xinan Hao. Infinitely many solutions for a class of biharmonic equations with indefinite potentials. AIMS Mathematics, 2020, 5(4): 3634-3645. doi: 10.3934/math.2020235

References

  • 1. Y. Chen, P. J. McKenna, Traveling waves in a nonlinearly suspension beam: Theoretical results and numerical observations, J. Differ. Equations, 135 (1997), 325-355.
  • 2. A. C. Lazer, P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, Siam. Rev., 32 (1990), 537-578.    
  • 3. P. J. McKenna, W. Walter, Traveling waves in a suspension bridge, Siam J. Appl. Math., 50 (1990), 703-715.    
  • 4. C. O. Alves, J. Marcos do Ó, O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal-Theor., 46 (2001), 121-133.    
  • 5. K. Kefi, K. Saoudi, On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions, Adv. Nonlinear Anal., 8 (2018), 1171-1183.    
  • 6. J. Liu, S. X. Chen, X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbb{R}^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
  • 7. A. Mao, W. Wang, Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in $\mathbb{R}^3$, J. Math. Anal. Appl., 459 (2018), 556-563.
  • 8. Y. Pu, X. P. Wu, C. L. Tang, Fourth-order Navier boundary value problem with combined nonlinearities, J. Math. Anal. Appl., 398 (2013), 798-813.    
  • 9. M. T. O. Pimenta, S. H. M. Soares, Singulary perturbed biharmonic problem with superlinear nonlinearities, Adv. Differential Equ., 19 (2014), 31-50.
  • 10. Y. Su, H. Chen, The existence of nontrivial solution for a class of sublinear biharmonic equations with steep potential well, Bound. Value Probl., 2018 (2018), 1-14.    
  • 11. X. Wang, A. Mao, A. Qian, High energy solutions of modified quasilinear fourth-order elliptic equation, Bound. Value Probl., 2018 (2018), 1-13.    
  • 12. Y. Wang, Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differ. Equations, 246 (2009), 3109-3125.    
  • 13. Y. Wei, Multiplicity results for some fourth-order elliptic equations, J. Math. Anal. Appl., 385 (2012), 797-807.    
  • 14. M. B. Yang, Z. F. Shen, Infinitely many solutions for a class of fourth order elliptic equations in $\mathbb{R}^N$, Acta Math. Sin., 24 (2008), 1269-1278.
  • 15. Y. W. Ye, C. L. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394 (2012), 841-854.    
  • 16. Y. W. Ye, C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbb{R}^N$, J. Math. Anal. Appl., 406 (2013), 335-351.
  • 17. Y. L. Yin, X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705.    
  • 18. J. Zhang, Z. Wei, Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems, Nonlinear Anal-Theor., 74 (2011), 7474-7485.    
  • 19. W. Zhang, X. H. Tang, J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwan. J. Math., 18 (2014), 645-659.    
  • 20. W. Zhang, X. H. Tang, J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368.    
  • 21. W. Zhang, X. H. Tang, J. Zhang, Existence and concentration of solutions for sublinear fourthorder elliptic equations, Electronic J. Differ. Eq., 2015 (2015), 1-9.    
  • 22. J. W. Zhou, X. Wu, Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl., 342 (2008), 542-558.    
  • 23. A. Bahrouni, H. Ounaies, V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, Racsam. Rev. R. Acad. A., 113 (2019), 1191-1210.
  • 24. A. Bahrouni, V. D. Rădulescu, D. Repovs, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.    
  • 25. G. Bonanno, G. D'Aguì, A. Sciammetta, Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions, Opuscula Math., 39 (2018), 159-174.
  • 26. Y. Li, D. B. Wang, J. Zhang, Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity, AIMS Math., 5 (2020), 2100-2112.    
  • 27. N. S. Papageorgiou, V. D. Radulescu, D. D. Repovs, Nonlinear analysis-theory and methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
  • 28. N. S. Papageorgiou, V. D. Radulescu, D. D. Repovs, Positive solutions for nonlinear parametric singular Dirichlet problems, Bull. Math. Sci., 9 (2019), 1950011.
  • 29. H. R. Quoirin, K. Umezu, An elliptic equation with an indefinite sublinear boundary condition, Adv. Nonlinear Anal., 8 (2019), 175-192.
  • 30. D. B. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501. Available from: https://doi.org/10.1063/1.5074163.
  • 31. D. B. Wang, T. Li, X. Hao, Least-energy sign-changing solutions for KirchhoffSchrödinger-Poisson systems in $\mathbb{R}^3$, Bound. Value Probl., 75 (2019). Available from: https://doi.org/10.1186/s13661-019-1183-3.
  • 32. D. B. Wang, Y. Ma, W. Guan, Least energy sign-changing solutions for the fractional Schrödinger-Poisson systems in $\mathbb{R}^3$, Bound. Value Probl., 25 (2019). Available from: https://doi.org/10.1186/s13661-019-1128-x.
  • 33. D. B. Wang, H. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301.    
  • 34. D. B. Wang, H. Zhang, Y. Ma, et al. Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system with potential vanishing at infinity, J. Appl. Math. Comput., 61 (2019), 611-634.    
  • 35. D. B. Wang, J. Zhang, Least energy sign-changing solutions of fractional Kirchhoff-SchrödingerPoisson system with critical growth, App. Math. Lett., 106 (2020), 106372.
  • 36. J. Zhao, X. Liu, Z. Feng, Quasilinear equations with indefinite nonlinearity, Adv. Nonlinear Anal., 8 (2018), 1235-1251.    
  • 37. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in. Math., 65, American Mathematical Society, Providence, RI, 1986.
  • 38. A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.    
  • 39. R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.    
  • 40. A. Bahrouni, H. Ounaies, V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potential, P. Roy. Soc. Edinburgh, Sect. A, 145 (2015), 445-465.    

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved