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Existence of multiple solutions for a fourth-order problem with variable exponent
1. | Department of Science and High Technology, University of Insubria, Via Valleggio, 11 - 22100 Como, Italy |
2. | Department of Science and High Technology, and Riemann International School of Mathematics, University of Insubria, Via G.B. Vico, 46 - 21100 Varese, Italy |
We provide a new multiplicity result for a weighted $ p(x) $-biharmonic problem on a bounded domain $ \Omega $ of $ \mathbb R^n $ with Navier conditions on $ \partial\Omega $. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in $ \Omega $.
References:
[1] |
S. Baraket and V. Rǎdulescu,
Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent, Adv. Nonlinear Stud., 16 (2016), 409-419.
doi: 10.1515/ans-2015-5032. |
[2] |
M. M. Boureanu, V. Rǎdulescu and D. Repovš,
On a $p(x)$-biharmonic problem with no-flux boundary condition, Comput. Math. Appl., 72 (2016), 2505-2515.
doi: 10.1016/j.camwa.2016.09.017. |
[3] |
F. Cammaroto and L. Vilasi,
Multiplicity results for a Neumann boundary value problem involving the $p(x)$-Laplacian, Taiwan. J. Math., 16 (2012), 621-634.
doi: 10.11650/twjm/1500406606. |
[4] |
F. Cammaroto and L. Vilasi,
Sequences of weak solutions for a Navier problem driven by the $p(x)$-biharmonic operator, Minimax Theory Appl., 4 (2019), 71-85.
doi: 10.1016/j.jmaa.2013.01.013. |
[5] |
Y. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[6] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math., vol. 2017, Springer-Verlag, Berlin, 2011.
doi: 10.1007/978-3-642-18363-8. |
[7] |
A. R. El Amrouss and A. Ourraoui,
Existence of solutions for a boundary problem involving $p(x)$-biharmonic operator, Bol. Soc. Parana. Mat., 31 (2013), 179-192.
doi: 10.5269/bspm.v31i1.15148. |
[8] |
X. L. Fan and D. Zhao,
On the spaces $L^{p(x)}$ and $W^{m, p(x)}$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[9] |
F. Faraci and A. Kristály,
Three non-zero solutions for a nonlinear eigenvalue problem, J. Math. Anal. Appl., 394 (2012), 225-230.
doi: 10.1016/j.jmaa.2012.04.045. |
[10] |
C. Farkas and I. I. Mezei,
Group-invariant multiple solutions for quasilinear elliptic problems on strip-like domains, Nonlinear Anal., 79 (2013), 238-246.
doi: 10.1016/j.na.2012.11.012. |
[11] |
T. C. Halsey,
Electrorheological fluids, Science, 258 (1992), 761-766.
|
[12] |
K. Kefi and V. Rǎdulescu, On a $p(x)$-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68 (2017), Paper No. 80, 13 pp.
doi: 10.1007/s00033-017-0827-3. |
[13] |
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. |
[14] |
O. Kováčik and J. Rákosník,
On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
|
[15] |
V. Rǎdulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis, Boca Raton (FL), 2015.
doi: 10.1201/b18601.![]() ![]() ![]() |
[16] |
B. Ricceri,
A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal., 11 (2010), 503-511.
|
[17] |
B. Ricceri,
A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.
doi: 10.1016/S0377-0427(99)00269-1. |
[18] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000. |
[19] |
L. Vilasi, A non-homogeneous elliptic problem in low dimensions with three symmetric solutions, J. Math. Anal. Appl., (2020), 124074.
doi: 10.1016/j.jmaa.2020.124074. |
[20] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.
|
show all references
References:
[1] |
S. Baraket and V. Rǎdulescu,
Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent, Adv. Nonlinear Stud., 16 (2016), 409-419.
doi: 10.1515/ans-2015-5032. |
[2] |
M. M. Boureanu, V. Rǎdulescu and D. Repovš,
On a $p(x)$-biharmonic problem with no-flux boundary condition, Comput. Math. Appl., 72 (2016), 2505-2515.
doi: 10.1016/j.camwa.2016.09.017. |
[3] |
F. Cammaroto and L. Vilasi,
Multiplicity results for a Neumann boundary value problem involving the $p(x)$-Laplacian, Taiwan. J. Math., 16 (2012), 621-634.
doi: 10.11650/twjm/1500406606. |
[4] |
F. Cammaroto and L. Vilasi,
Sequences of weak solutions for a Navier problem driven by the $p(x)$-biharmonic operator, Minimax Theory Appl., 4 (2019), 71-85.
doi: 10.1016/j.jmaa.2013.01.013. |
[5] |
Y. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[6] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math., vol. 2017, Springer-Verlag, Berlin, 2011.
doi: 10.1007/978-3-642-18363-8. |
[7] |
A. R. El Amrouss and A. Ourraoui,
Existence of solutions for a boundary problem involving $p(x)$-biharmonic operator, Bol. Soc. Parana. Mat., 31 (2013), 179-192.
doi: 10.5269/bspm.v31i1.15148. |
[8] |
X. L. Fan and D. Zhao,
On the spaces $L^{p(x)}$ and $W^{m, p(x)}$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[9] |
F. Faraci and A. Kristály,
Three non-zero solutions for a nonlinear eigenvalue problem, J. Math. Anal. Appl., 394 (2012), 225-230.
doi: 10.1016/j.jmaa.2012.04.045. |
[10] |
C. Farkas and I. I. Mezei,
Group-invariant multiple solutions for quasilinear elliptic problems on strip-like domains, Nonlinear Anal., 79 (2013), 238-246.
doi: 10.1016/j.na.2012.11.012. |
[11] |
T. C. Halsey,
Electrorheological fluids, Science, 258 (1992), 761-766.
|
[12] |
K. Kefi and V. Rǎdulescu, On a $p(x)$-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68 (2017), Paper No. 80, 13 pp.
doi: 10.1007/s00033-017-0827-3. |
[13] |
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. |
[14] |
O. Kováčik and J. Rákosník,
On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
|
[15] |
V. Rǎdulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis, Boca Raton (FL), 2015.
doi: 10.1201/b18601.![]() ![]() ![]() |
[16] |
B. Ricceri,
A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal., 11 (2010), 503-511.
|
[17] |
B. Ricceri,
A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.
doi: 10.1016/S0377-0427(99)00269-1. |
[18] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000. |
[19] |
L. Vilasi, A non-homogeneous elliptic problem in low dimensions with three symmetric solutions, J. Math. Anal. Appl., (2020), 124074.
doi: 10.1016/j.jmaa.2020.124074. |
[20] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.
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