Keywords
Fourth order elliptic equation
Constraint variational method
Concave-convex nonlinearities
Quantitative deformation lemma
How to Cite
Abstract
In this paper, we study the following fourth order elliptic equation:
Δ²u - Δu + V(x)u = κ(x)|u|q-2u + |u|p-2u in RN,
where Δ² := Δ(Δ) is the biharmonic operator, 4 > N, 1 < q < 2 < p < 2* := 2N / (N - 4). Assuming that V(x) satisfies a class of coercive conditions and the nonnegative weighted function κ(x) belongs to Lp / (p-q)(RN), we obtain the existence of one sign-changing solution with the help of constraint variational method and quantitative deformation lemma.
The novelty of this paper is that when the nonlinearity is the combination of concave and convex functions, we are able to obtain the existence of sign-changing solutions. Some recent results are improved and generalized significantly.
References
Lazer AC, Mckenna PJ. Global bifurcation and a theorem of Tarantello. J Math Anal Appl. 1994; 181: 648-55. https://doi.org/10.1006/jmaa.1994.1049
Lazer AC, Mckenna PJ. Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 1990; 32: 537-78. https://doi.org/10.1137/1032120
Chen Y, Mckenna PJ. Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations. J Differ Equ. 1997; 135: 325-55. https://doi.org/10.1006/jdeq.1996.3155
Pao CV. On fourth-order elliptic boundary value problems. Proc Am Math Soc. 2000; 128: 1023-30. https://doi.org/10.1090/S0002-9939-99-05430-1
Tarantella G. A note on a semilinear elliptic problem. Differ Integral Equ. 1992; 5: 561-5. https://doi.org/10.57262/die/1370979319
Ahmed NU, Harbi H. Mathematical analysis of dynamic models of suspension bridges. SIAM J Appl Math. 1998; 58: 853-74. https://doi.org/10.1137/S0036139996308698
Han YZ, Cao CL, Li, JK. Multiplicity of weak solutions to a fourth-order elliptic equation with sign-changing weight function. Complex Var Elliptic Equ. 2020; 65: 1462-73. https://doi.org/10.1080/17476933.2019.1662405
Li C, Agarwal RP, Ou, ZQ. Existence of three nontrivial solutions for a class of fourth-order elliptic equations. Topol Methods Nonlinear Anal. 2018; 52: 331-44. https://doi.org/10.12775/TMNA.2018.001
Pei RC, Xia, HM. Multiplicity results for some fourth-order elliptic equations with combined nonlinearities. Aims Math. 2023; 6: 14704-25. https://doi.org/10.3934/math.2023752
Wang FL, An, YK. Existence and multiplicity of solutions for a fourth-order elliptic equation. Bound Value Probl. 2012; 6: 1-9. https://doi.org/10.1186/1687-2770-2012-6
Wang WH, Zang AB, Zhao PH. Multiplicity of solutions for a class of fourth-order elliptical equations, Nonlinear Anal. 2009; 70: 4377-85. https://doi.org/10.1016/j.na.2008.10.020
Wei YH. Multiplicity results for some fourth-order elliptic equations. J Math Anal Appl. 2012; 385: 797-807. https://doi.org/10.1016/j.jmaa.2011.07.011
Zhang J, Wei ZL. Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems. Nonlinear Anal. 2011; 74: 7474-85. https://doi.org/10.1016/j.na.2011.07.067
Zuo JB, El Allali Z, Taarabti S. Nontrivial solutions of a class of fourth-order elliptic problems with potentials. Fractal fract. 2022; 6: 1-12. https://doi.org/10.3390/fractalfract6100568
Cheng BT. High energy solutions for the fourth-order elliptic equations in R^N Bound. Value Probl. 2014; 199: 1-11. https://doi.org/10.1186/s13661-014-0199-y
Liu J, Chen SX, Wu X. Existence and multiplicity of solutions for a class of fourth-order elliptic equations in R^N. J Math Anal Appl. 2012; 395: 608-15. https://doi.org/10.1016/j.jmaa.2012.05.063
Su Y, Chen HB. The existence of nontrivial solution for biharmonic equation with sign-changing potential. Math Methods Appl Sci. 2018; 41: 6170-83. https://doi.org/10.1002/mma.5127
Wu ZJ, Chen HB. Fourth order elliptic equation involving sign-changing weight function in R^N. J Dyn Control Syst. 2022; 30: 830-49. https://doi.org/10.1007/s10883-023-09647-z
Wu ZJ, Chen HB. A class of fourth-order elliptic equations with concave and convex nonlinearities in R^N. Electron J Qual Theory Differ Equ. 2021; 71: 1-16. https://doi.org/10.14232/ejqtde.2021.1.71
Yang MB, Shen, ZF. Infinitely many solutions for a class of fourth order elliptic equations in R^N. Acta Math Sin Engl Ser. 2008; 24: 1269-78. https://doi.org/10.1007/s10114-008-5423-1
Ye YW, Tang CL. Existence and multiplicity of solutions for fourth-order elliptic equations in R^N. J Math Anal Appl. 2013; 406: 335-51. https://doi.org/10.1016/j.jmaa.2013.04.079
Ye YW, Tang CL. Infinitely many solutions for fourth-order elliptic equations. J Math Anal Appl 2012; 394: 841-54. https://doi.org/10.1016/j.jmaa.2012.04.041
Yin YL, Wu X. High energy solutions and nontrival solutions for fourth-order elliptic equations. J Math Anal Appl. 2011; 375: 699-705. https://doi.org/10.1016/j.jmaa.2010.10.019
Zhang HS, Li TX, Wu TF. Existence and multiplicity of nontrivial solutions for biharmonic equations with singular weight functions. Appl Math Lett. 2020; 105: 1-8. https://doi.org/10.1016/j.aml.2020.106335
Zhang W, Tang XH, Zhang J. Infinitely many solutions for fourth-order elliptic equations with sign-changing potential. Taiwan J Math. 2014; 18: 645-59. https://doi.org/ 10.11650/tjm.18.2014.3584
Zhang W, Tang XH, Zhang J. Infinitely many solutions for fourth-order elliptic equations with general potentials. J Math Anal Appl. 2013; 407: 359-68. https://doi.org/10.1016/j.jmaa.2013.05.044
Zhang W, Tang XH, Zhang J, Luo ZM. Multiplicity and concentration of solutions for fourth-order elliptic equations with mixed nonlinearity. Electron J Differential Equations. 2017; 2017: 1-15.
Zhang W, Zhang J, Luo ZM. Multiple solutions for the fourth-order elliptic equation with vanishing potential. Appl Math Lett. 2017; 73: 98-105. https://doi.org/10.1016/j.aml.2017.04.030
Che G, Chen HB. Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations. J Appl Math Comput. 2017; 53: 33-49. https://doi.org/10.1007/s12190-015-0956-9
Pimenta MT. Radial sign-changing solutions to biharmonic nonlinear Schrödinger equations. Bound. Value Probl. 2015; 21: 1-17. https://doi.org/10.1186/s13661-015-0282-z
Guan W, Zhang HB. Sign-changing solutions for Schrödinger-Kirchhoff-type fourth-order equation with potential vanishing at infinity. J Inequal Appl. 2021; 2021: 1-22. https://doi.org/10.1186/s13660-021-02552-8
Khoutir S, Chen HB. Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in R^N. J Appl Math Comput. 2017; 55: 25-39. https://doi.org/10.1007/s12190-016-1023-x
Liu HL, Xiao QZ, Shi HX, Chen HB, Liu ZS. Ground state and nodal solutions for a class of biharmonic equations with singular potentials. J Appl Anal Comput. 2019; 9: 1393-406. https://doi.org/10.11948/2156-907X.20180253
Pu HL, Li SQ, Liang SH, Repovs DD. Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N. Electron J Differential Equations. 2021; 2021: 1-20.
Zhang HB, Guan W. Least energy sign-changing solutions for fourth-order Kirchhoff-type equation with potential vanishing at infinity. J Appl Math Comput. 2020; 64: 157-77. https://doi.org/10.1007/s12190-020-01349-0
Zhang W, Tang XH, Cheng BT, Zhang J. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Commun Pure Appl Anal. 2016; 15: 2161-77. https://doi.org/10.3934/cpaa.2016032
Zhang ZH. Sign-changing solutions for a class of Schröodinger-Bopp-Podolsky system with concave-convex nonlinearities. J Math Anal Appl. 2024; 530(1): 127712. https://doi.org/10.1016/j.jmaa.2023.127712
Chen SX, Liu J, Wu X. Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on R^N. Appl Math Comput. 2014; 248: 593-601. https://doi.org/10.1016/j.amc.2014.10.021
Willem M. Minimax theorems. Boston: Birkhäuser Boston Inc.; 1996.
Zou WM, Schechterc M. Critical point theory and its applications. Springer; 2006.
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