Existence of multiple solutions for a fourth-order problem with variable exponent

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May 2022, 27(5): 2471-2481. doi: 10.3934/dcdsb.2021141

Existence of multiple solutions for a fourth-order problem with variable exponent

Marco Donatelli ^1, and Luca Vilasi ^2,,

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

^* Corresponding author

Received December 2020 Revised March 2021 Published May 2022 Early access May 2021

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 $.

Keywords: p(x)-biharmonic operator, Navier conditions, multiplicity, variational methods.

Mathematics Subject Classification: Primary: 35A01; Secondary: 35G30.

Citation: Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2471-2481. doi: 10.3934/dcdsb.2021141

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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