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

On principal eigenvalues of biharmonic systems

Lingju Kong , and  Roger Nichols

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.

Keywords: Biharmonic system, principal eigenvalues, eigenpair, Picone's identity, isolated.
Mathematics Subject Classification: Primary:35P30, 35J48.
Citation: Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & 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.  Google Scholar

[2]

J. Barrow, R. DeYeso Ⅲ, L. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Gesztesy, M. 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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Zhang, Y. 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.  Google Scholar

[29]

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

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.  Google Scholar

[2]

J. Barrow, R. DeYeso Ⅲ, L. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Gesztesy, M. 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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Zhang, Y. 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.  Google Scholar

[29]

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

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