# American Institute of Mathematical Sciences

January  2020, 19(1): 391-405. doi: 10.3934/cpaa.2020020

## Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion

 a. Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China b. Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China

* Corresponding author

Received  December 2018 Revised  May 2019 Published  July 2019

Fund Project: Research are supported by the Hunan Province Natural Science Foundation of China (Grant No. 2017JJ3222) and the China Postdoctoral Science Foundation funded project (Grant No. 2018M643039)

This paper is concerned with a class of biharmonic elliptic differential inclusion in $\mathbb R^N$. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals.

Citation: Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020
##### References:
 [1] I. Abrahams and A. Davis, Deflection of a partially clamped elastic plate, in: IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity, Fluid Mechanics and Its Application, 68 (2002), 303–312.Google Scholar [2] C. Alves, A variational approach to discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 265 (2002), 103-127. doi: 10.1006/jmaa.2001.7698. Google Scholar [3] C. Alves and G. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbb R^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. Google Scholar [4] C. O. Alves and S. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbb R^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033. Google Scholar [5] C. O. Alves and M. Souto, Multiplicity of positive solutions for a class of problems with exponential critical growth in $\mathbb R^2$, J. Differential Equations, 244 (2008), 1502-1520. doi: 10.1016/j.jde.2007.09.007. Google Scholar [6] T. Bartsh and Z. Wang, Existence and multiplicity results for some superlinear elliptic problem on $\mathbb R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. Google Scholar [7] D. Biles and J. Spraker, Existence of positive solutions for a fourth order differential inclusion, Differ. Equ. Appl., 4 (2012), 539-546. doi: 10.7153/dea-04-31. Google Scholar [8] A. Cernea, Existence of solutions for a certain boundary value problem associated to a fourth order differential inclusion, Int. J. Math. Anal., 14 (2017), 27–33.Google Scholar [9] J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $\mathbb R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar [10] K. Chang, Variational methods for nondifferentiabe functionals and their applications to partial differential inequalities, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0. Google Scholar [11] B. Cheng and X Tang, High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential, Comput. Math. Appl., 73 (2017), 27-36. doi: 10.1016/j.camwa.2016.10.015. Google Scholar [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Google Scholar [13] L. Gasiński and N. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC Press, Boca Raton, FL, 2005. Google Scholar [14] A. Lazer and P. Mckenna, Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120. Google Scholar [15] S. Liang and J. Zhang, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in $\mathbb R^N$, J. Math. Phy., 57 (2016), 1-13. doi: 10.1063/1.4967976. Google Scholar [16] J. Liu, S. Chen and 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. doi: 10.1016/j.jmaa.2012.05.063. Google Scholar [17] P. Mckenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041. Google Scholar [18] M. Pimenta and S. Soares, Existence and concentration of soltions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289. doi: 10.1016/j.jmaa.2012.01.039. Google Scholar [19] J. Sun and T. F. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115. doi: 10.1016/j.na.2014.11.009. Google Scholar [20] Y. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019. Google Scholar [21] J. Zhang and Y. Zhou, Existence of a nontrivial solutions for a class of hemivariational inequality problems at double resonance, Nonlinear Anal., 74 (2011), 4319-4329. doi: 10.1016/j.na.2011.02.038. Google Scholar [22] W. Zhang, H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368. doi: 10.1016/j.jmaa.2013.05.044. Google Scholar [23] W. Zhang, X. Tang, B. Cheng and J. Zhang, Sign-changing solutions for fourth elliptic equations with Kirchhoff-type, Comm. Pure Appl. Anal., 15 (2016), 2161-2177. doi: 10.3934/cpaa.2016032. Google Scholar [24] W. Zhang, J. Zhang and Z. Luo, Multiple solutions for the forth-order elliptic equations with vanishing potential, Appl. Math. Letters, 73 (2017), 98-105. doi: 10.1016/j.aml.2017.04.030. Google Scholar [25] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006. Google Scholar

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##### References:
 [1] I. Abrahams and A. Davis, Deflection of a partially clamped elastic plate, in: IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity, Fluid Mechanics and Its Application, 68 (2002), 303–312.Google Scholar [2] C. Alves, A variational approach to discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 265 (2002), 103-127. doi: 10.1006/jmaa.2001.7698. Google Scholar [3] C. Alves and G. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbb R^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. Google Scholar [4] C. O. Alves and S. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbb R^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033. Google Scholar [5] C. O. Alves and M. Souto, Multiplicity of positive solutions for a class of problems with exponential critical growth in $\mathbb R^2$, J. Differential Equations, 244 (2008), 1502-1520. doi: 10.1016/j.jde.2007.09.007. Google Scholar [6] T. Bartsh and Z. Wang, Existence and multiplicity results for some superlinear elliptic problem on $\mathbb R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. Google Scholar [7] D. Biles and J. Spraker, Existence of positive solutions for a fourth order differential inclusion, Differ. Equ. Appl., 4 (2012), 539-546. doi: 10.7153/dea-04-31. Google Scholar [8] A. Cernea, Existence of solutions for a certain boundary value problem associated to a fourth order differential inclusion, Int. J. Math. Anal., 14 (2017), 27–33.Google Scholar [9] J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $\mathbb R^N$, Nonlinear Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar [10] K. Chang, Variational methods for nondifferentiabe functionals and their applications to partial differential inequalities, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0. Google Scholar [11] B. Cheng and X Tang, High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential, Comput. Math. Appl., 73 (2017), 27-36. doi: 10.1016/j.camwa.2016.10.015. Google Scholar [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Google Scholar [13] L. Gasiński and N. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC Press, Boca Raton, FL, 2005. Google Scholar [14] A. Lazer and P. Mckenna, Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120. Google Scholar [15] S. Liang and J. Zhang, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in $\mathbb R^N$, J. Math. Phy., 57 (2016), 1-13. doi: 10.1063/1.4967976. Google Scholar [16] J. Liu, S. Chen and 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. doi: 10.1016/j.jmaa.2012.05.063. Google Scholar [17] P. Mckenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041. Google Scholar [18] M. Pimenta and S. Soares, Existence and concentration of soltions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289. doi: 10.1016/j.jmaa.2012.01.039. Google Scholar [19] J. Sun and T. F. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115. doi: 10.1016/j.na.2014.11.009. Google Scholar [20] Y. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019. Google Scholar [21] J. Zhang and Y. Zhou, Existence of a nontrivial solutions for a class of hemivariational inequality problems at double resonance, Nonlinear Anal., 74 (2011), 4319-4329. doi: 10.1016/j.na.2011.02.038. Google Scholar [22] W. Zhang, H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368. doi: 10.1016/j.jmaa.2013.05.044. Google Scholar [23] W. Zhang, X. Tang, B. Cheng and J. Zhang, Sign-changing solutions for fourth elliptic equations with Kirchhoff-type, Comm. Pure Appl. Anal., 15 (2016), 2161-2177. doi: 10.3934/cpaa.2016032. Google Scholar [24] W. Zhang, J. Zhang and Z. Luo, Multiple solutions for the forth-order elliptic equations with vanishing potential, Appl. Math. Letters, 73 (2017), 98-105. doi: 10.1016/j.aml.2017.04.030. Google Scholar [25] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006. Google Scholar
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