January  2021, 20(1): 1-15. doi: 10.3934/cpaa.2020254

On principal eigenvalues of biharmonic systems

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

* Corresponding author

Received  June 2020 Revised  July 2020 Published  January 2021 Early access  October 2020

We prove the existence, positivity, simplicity, uniqueness up to nonnegative eigenfunctions, and isolation of the principle eigenvalue of a biharmonic system. We also provide the extension of our results to a related system.

Citation: Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure and Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254
References:
[1]

A. Ayoujil, Existence and nonexistence results for weighted fourth order eigenvalue problems with variable exponent, Bol. Soc. Parana. Math. (3), 37 (2019), 55-66.  doi: 10.5269/bspm.v37i3.31657.

[2]

J. BarrowR. DeYeso ⅢL. Kong and and F. Petronella, Positive radially symmetric solutions for a system of quasilinear biharmonic equation in the plane, Electron. J. Differ. Equ., 2015 (2015), 1-11. 

[3]

J. Benedikt and P. Drábek, Estimates of the principal eigenvalue of the $p$-biharmonic operator, Nonlinear Anal., 75 (2012), 5374-5379.  doi: 10.1016/j.na.2012.04.055.

[4]

J. Benedikt and P. Drábek, Asymptotics for the principal eigenvalue of the $p$-biharmonic operator on the ball as $p$ approaches $1$, Nonlinear Anal., 95 (2014), 735-742.  doi: 10.1016/j.na.2013.10.016.

[5] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. 
[6]

Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations, J. Differ. Equ., 136 (1997), 325-355. doi: 10.1006/jdeq.1996.3155.

[7]

A. L. A. de Araujo and L. F. O. Faria, Existence of solutions to biharmonic systems with singular nonlinearity, Electron. J. Differ. Equ., 2016 (2016), 1-12. 

[8]

R. Demarque and N. D. H. Lisboa, Radial solutions for inhomogeneous biharmonic elliptic systems, Electron. J. Differ. Equ., 2018 (2018), 1-14. 

[9]

Z. Deng and Y. Huang, Multiple symmetric results for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in $ {\mathbb R}^N$, Acta Math. Sci., 37B (2017), 1665-1684.  doi: 10.1016/S0252-9602(17)30099-1.

[10]

P. Drábek and M. Ótani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Diferr. Equ., 2001 (2001), 1-19. 

[11]

G. Dwivedi, Picone identity for $p$-biharmonic operator and its applications, arXiv: 1503.05535v2.

[12]

W. Faris, Quadratic forms and essential self-adjointness, Helv. Phys. Acta, 45 (1972/73), 1074-1088. 

[13]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019.

[14]

B. Ge, Q. Zhou, and Y. Wu, Eigenvalues of the $p(x)$-biharmonic operator with indefinite weight, Z. Angew. Math. Phys., 66 (2015), 1007-1021. doi: 10.1007/s00033-014-0465-y.

[15]

F. GesztesyM. Mitrea and and R. Nichols, Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions, J. Anal. Math., 122 (2014), 229-287.  doi: 10.1007/s11854-014-0008-7.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second order, 2$^nd$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.

[17]

J. Jaroš, Picone's identity for the $p$-biharmonic operator with applications, Electron. J. Differ. Equ., 2011 (2011), 1-6. 

[18]

D. Kang and C. Kao, Minimization of inhomogeneous biharmonic eigenvalue problems, Appl. Math. Model., 51 (2017), 587-604.  doi: 10.1016/j.apm.2017.07.015.

[19]

D. Kang and P. Xiong, Existence and nonexistence results for critical biharmonic systems involving multiple singularities, J. Math. Anal. Appl., 452 (2017), 469-487.  doi: 10.1016/j.jmaa.2017.03.011.

[20]

L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc., 143 (2015), 249–258. doi: 10.1090/S0002-9939-2014-12213-1.

[21]

L. Kong, Multiple solutions for fourth order elliptic problems with $p(x)$-biharmonic operators, Opuscula Math., 36 (2016), 252-264.  doi: 10.7494/OpMath.2016.36.2.253.

[22]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connection with nonlinear analysis, it SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120.

[23]

L. Lin and S. Heidarkhani, Existence of three solutions to a double eigenvalue problem for the $p$-harmonic equation, Ann. Polon. Math., 104 (2012), 71-80.  doi: 10.4064/ap104-1-5.

[24]

L. Lin and C. Tang, Existence of three solutions for $(p,q)$-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805.  doi: 10.1016/j.na.2010.04.018.

[25]

P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.  doi: 10.1137/0150041.

[26]

Y. Sang and Y. Ren, A critical $p$-biharmonic system with negative exponents, Comput. Math. Appl., 79 (2020), 1335-1361.  doi: 10.1016/j.camwa.2019.08.032.

[27]

G. Savaré, On the regularity of the positive part of functions, Nonlinear Anal., 27 (1996), 1055-1074.  doi: 10.1016/0362-546X(95)00104-4.

[28]

S. ZhangY. Xi and X. Ji, A multi-level mixed element method for the eigenvalue problem of biharmonic equation, J. Sci. Comput., 75 (2018), 1415-1444.  doi: 10.1007/s10915-017-0592-7.

[29]

N. B. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr., 281 (2008), 1351-1365.  doi: 10.1002/mana.200510683.

show all references

References:
[1]

A. Ayoujil, Existence and nonexistence results for weighted fourth order eigenvalue problems with variable exponent, Bol. Soc. Parana. Math. (3), 37 (2019), 55-66.  doi: 10.5269/bspm.v37i3.31657.

[2]

J. BarrowR. DeYeso ⅢL. Kong and and F. Petronella, Positive radially symmetric solutions for a system of quasilinear biharmonic equation in the plane, Electron. J. Differ. Equ., 2015 (2015), 1-11. 

[3]

J. Benedikt and P. Drábek, Estimates of the principal eigenvalue of the $p$-biharmonic operator, Nonlinear Anal., 75 (2012), 5374-5379.  doi: 10.1016/j.na.2012.04.055.

[4]

J. Benedikt and P. Drábek, Asymptotics for the principal eigenvalue of the $p$-biharmonic operator on the ball as $p$ approaches $1$, Nonlinear Anal., 95 (2014), 735-742.  doi: 10.1016/j.na.2013.10.016.

[5] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. 
[6]

Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations, J. Differ. Equ., 136 (1997), 325-355. doi: 10.1006/jdeq.1996.3155.

[7]

A. L. A. de Araujo and L. F. O. Faria, Existence of solutions to biharmonic systems with singular nonlinearity, Electron. J. Differ. Equ., 2016 (2016), 1-12. 

[8]

R. Demarque and N. D. H. Lisboa, Radial solutions for inhomogeneous biharmonic elliptic systems, Electron. J. Differ. Equ., 2018 (2018), 1-14. 

[9]

Z. Deng and Y. Huang, Multiple symmetric results for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in $ {\mathbb R}^N$, Acta Math. Sci., 37B (2017), 1665-1684.  doi: 10.1016/S0252-9602(17)30099-1.

[10]

P. Drábek and M. Ótani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Diferr. Equ., 2001 (2001), 1-19. 

[11]

G. Dwivedi, Picone identity for $p$-biharmonic operator and its applications, arXiv: 1503.05535v2.

[12]

W. Faris, Quadratic forms and essential self-adjointness, Helv. Phys. Acta, 45 (1972/73), 1074-1088. 

[13]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019.

[14]

B. Ge, Q. Zhou, and Y. Wu, Eigenvalues of the $p(x)$-biharmonic operator with indefinite weight, Z. Angew. Math. Phys., 66 (2015), 1007-1021. doi: 10.1007/s00033-014-0465-y.

[15]

F. GesztesyM. Mitrea and and R. Nichols, Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions, J. Anal. Math., 122 (2014), 229-287.  doi: 10.1007/s11854-014-0008-7.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second order, 2$^nd$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.

[17]

J. Jaroš, Picone's identity for the $p$-biharmonic operator with applications, Electron. J. Differ. Equ., 2011 (2011), 1-6. 

[18]

D. Kang and C. Kao, Minimization of inhomogeneous biharmonic eigenvalue problems, Appl. Math. Model., 51 (2017), 587-604.  doi: 10.1016/j.apm.2017.07.015.

[19]

D. Kang and P. Xiong, Existence and nonexistence results for critical biharmonic systems involving multiple singularities, J. Math. Anal. Appl., 452 (2017), 469-487.  doi: 10.1016/j.jmaa.2017.03.011.

[20]

L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc., 143 (2015), 249–258. doi: 10.1090/S0002-9939-2014-12213-1.

[21]

L. Kong, Multiple solutions for fourth order elliptic problems with $p(x)$-biharmonic operators, Opuscula Math., 36 (2016), 252-264.  doi: 10.7494/OpMath.2016.36.2.253.

[22]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connection with nonlinear analysis, it SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120.

[23]

L. Lin and S. Heidarkhani, Existence of three solutions to a double eigenvalue problem for the $p$-harmonic equation, Ann. Polon. Math., 104 (2012), 71-80.  doi: 10.4064/ap104-1-5.

[24]

L. Lin and C. Tang, Existence of three solutions for $(p,q)$-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805.  doi: 10.1016/j.na.2010.04.018.

[25]

P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.  doi: 10.1137/0150041.

[26]

Y. Sang and Y. Ren, A critical $p$-biharmonic system with negative exponents, Comput. Math. Appl., 79 (2020), 1335-1361.  doi: 10.1016/j.camwa.2019.08.032.

[27]

G. Savaré, On the regularity of the positive part of functions, Nonlinear Anal., 27 (1996), 1055-1074.  doi: 10.1016/0362-546X(95)00104-4.

[28]

S. ZhangY. Xi and X. Ji, A multi-level mixed element method for the eigenvalue problem of biharmonic equation, J. Sci. Comput., 75 (2018), 1415-1444.  doi: 10.1007/s10915-017-0592-7.

[29]

N. B. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr., 281 (2008), 1351-1365.  doi: 10.1002/mana.200510683.

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