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On principal eigenvalues of biharmonic systems
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA |
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.
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. 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.
|
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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.
|
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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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