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  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 & 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. 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.   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. 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.  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. 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.  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. 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.   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. 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.  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. 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.  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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