• Previous Article
    On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$
  • DCDS-S Home
  • This Issue
  • Next Article
    Classical solutions for the system $\bf {\text{curl}\, v = g}$, with vanishing Dirichlet boundary conditions
April  2019, 12(2): 231-243. doi: 10.3934/dcdss.2019016

Fourth-order problems with Leray-Lions type operators in variable exponent spaces

Department of Applied Mathematics, University of Craiova, 200585 Craiova, Rumania

Received  May 2017 Revised  November 2017 Published  August 2018

The Leray-Lions operators are versatile enough to be particularized to various elliptic operators, so they receive a lot of attention. This paper introduces to the mathematical literature Leray-Lions type operators that are appropriate for the study of the variable exponent problems of higher order. We establish some properties concerning these general operators and then we apply them to a fourth order problem with variable exponents.

Citation: Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016
References:
[1]

G. A. AfrouziM. Mirzapour and N. T. Chung, Existence and non-existence of solutions for a $p(x)$-biharmonic problem, Electronic Journal of Differential Equations, 2015 (2015), 1-8. 

[2]

A. Ayoujil and A. R. El Amrouss, Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differential Equations, 2011 (2011), 1-12. 

[3]

A. Ayoujil and A. R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal., 71 (2009), 4916-4926.  doi: 10.1016/j.na.2009.03.074.

[4]

M.-M. Boureanu, A new class of nonhomogeneous differential operator and applications to anisotropic systems, Complex Variables and Elliptic Equations, 61 (2016), 712-730.  doi: 10.1080/17476933.2015.1114614.

[5]

M.-M. BoureanuA. Matei and M. Sofonea, Nonlinear problems with $p(·)$-growth conditions and applications to antiplane contact models, Adv. Nonl. Studies, 14 (2014), 295-313.  doi: 10.1515/ans-2014-0203.

[6]

M.-M. BoureanuV. Rădulescu and D. Repovš, On a $p(·)$-biharmonic problem with no-flux boundary condition, Computers and Mathematics with Applications, 72 (2016), 2505-2515.  doi: 10.1016/j.camwa.2016.09.017.

[7]

M.-M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with $p(·)$ - growth conditions, Nonl. Anal. RWA, 14 (2013), 1829-1844.  doi: 10.1016/j.nonrwa.2012.12.001.

[8]

Y. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1386-1406.  doi: 10.1137/050624522.

[9]

F. ColasuonnoP. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving $p$-Laplacian type operators, Nonlinear Anal., 75 (2012), 4496-4512.  doi: 10.1016/j.na.2011.09.048.

[10]

D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4536-6.

[11]

D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer Basel, 2013. doi: 10.1007/978-3-0348-0548-3.

[12]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.

[13]

C.-P. Dăneţ, Two maximum principles for a nonlinear fourth order equation from thin plate theory, Electronic J. Qualitative Theory of Diff. Eq., 31 (2014), 1-9. 

[14]

L. Diening, Maximal function on generalized Lebesgue spaces Lp(·), Mathematical Inequalities and Applications, 7 (2004), 245-253.  doi: 10.7153/mia-07-27.

[15]

A. El AmroussF. Moradi and M. Moussaoui, Existence of solutions for fourth-order PDEs with variable exponents, Electron. J. Differ. Equ., 2009 (2009), 1-13. 

[16]

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.

[17]

X. L. Fan, Solutions for $p(x)$-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312 (2005), 464-477.  doi: 10.1016/j.jmaa.2005.03.057.

[18]

X. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in ${\mathbb R}^N$, Nonlinear Anal., 59 (2004), 173-188.  doi: 10.1016/j.na.2004.07.009.

[19]

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.

[20]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. 

[21]

A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005.

[22]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321.  doi: 10.1016/j.na.2009.01.211.

[23]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France, 93 (1965), 97-107. 

[24]

L. Li and C. L. Tang, Existence and multiplicity of solutions for a class of $p(x)$-biharmonic equations, Acta Mathematica Scientia, 33 (2013), 155-170.  doi: 10.1016/S0252-9602(12)60202-1.

[25]

G. Molica BisciV. Radulescu and R. Servadei, Competition phenomena for elliptic equations involving a general operator in divergence form, Anal. Appl., 15 (2017), 51-82.  doi: 10.1142/S0219530515500116.

[26]

G. Molica Bisci and D. Repovš, Multiple solutions of $p$-biharmonic equations with Navier boundary conditions, Complex Variables and Elliptic Equations, 59 (2014), 271-284.  doi: 10.1080/17476933.2012.734301.

[27]

T. G. Myers, Thin films with high surface tension, SIAM Review, 40 (1998), 441-462.  doi: 10.1137/S003614459529284X.

[28]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital. Series IX, 3 (2010), 543-582. 

[29]

V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Quantitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601.

[30]

M. R${\rm{\dot{u}}}$žiĄka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer–Verlag Berlin, 2002.

[31]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738.  doi: 10.1016/j.jmaa.2008.07.068.

[32]

A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. T.M.A., 69 (2008), 3629-3636.  doi: 10.1016/j.na.2007.10.001.

[33]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66. 

show all references

References:
[1]

G. A. AfrouziM. Mirzapour and N. T. Chung, Existence and non-existence of solutions for a $p(x)$-biharmonic problem, Electronic Journal of Differential Equations, 2015 (2015), 1-8. 

[2]

A. Ayoujil and A. R. El Amrouss, Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differential Equations, 2011 (2011), 1-12. 

[3]

A. Ayoujil and A. R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal., 71 (2009), 4916-4926.  doi: 10.1016/j.na.2009.03.074.

[4]

M.-M. Boureanu, A new class of nonhomogeneous differential operator and applications to anisotropic systems, Complex Variables and Elliptic Equations, 61 (2016), 712-730.  doi: 10.1080/17476933.2015.1114614.

[5]

M.-M. BoureanuA. Matei and M. Sofonea, Nonlinear problems with $p(·)$-growth conditions and applications to antiplane contact models, Adv. Nonl. Studies, 14 (2014), 295-313.  doi: 10.1515/ans-2014-0203.

[6]

M.-M. BoureanuV. Rădulescu and D. Repovš, On a $p(·)$-biharmonic problem with no-flux boundary condition, Computers and Mathematics with Applications, 72 (2016), 2505-2515.  doi: 10.1016/j.camwa.2016.09.017.

[7]

M.-M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with $p(·)$ - growth conditions, Nonl. Anal. RWA, 14 (2013), 1829-1844.  doi: 10.1016/j.nonrwa.2012.12.001.

[8]

Y. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1386-1406.  doi: 10.1137/050624522.

[9]

F. ColasuonnoP. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving $p$-Laplacian type operators, Nonlinear Anal., 75 (2012), 4496-4512.  doi: 10.1016/j.na.2011.09.048.

[10]

D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4536-6.

[11]

D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer Basel, 2013. doi: 10.1007/978-3-0348-0548-3.

[12]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.

[13]

C.-P. Dăneţ, Two maximum principles for a nonlinear fourth order equation from thin plate theory, Electronic J. Qualitative Theory of Diff. Eq., 31 (2014), 1-9. 

[14]

L. Diening, Maximal function on generalized Lebesgue spaces Lp(·), Mathematical Inequalities and Applications, 7 (2004), 245-253.  doi: 10.7153/mia-07-27.

[15]

A. El AmroussF. Moradi and M. Moussaoui, Existence of solutions for fourth-order PDEs with variable exponents, Electron. J. Differ. Equ., 2009 (2009), 1-13. 

[16]

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.

[17]

X. L. Fan, Solutions for $p(x)$-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312 (2005), 464-477.  doi: 10.1016/j.jmaa.2005.03.057.

[18]

X. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in ${\mathbb R}^N$, Nonlinear Anal., 59 (2004), 173-188.  doi: 10.1016/j.na.2004.07.009.

[19]

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.

[20]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618. 

[21]

A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser Verlag, 2005.

[22]

V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321.  doi: 10.1016/j.na.2009.01.211.

[23]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France, 93 (1965), 97-107. 

[24]

L. Li and C. L. Tang, Existence and multiplicity of solutions for a class of $p(x)$-biharmonic equations, Acta Mathematica Scientia, 33 (2013), 155-170.  doi: 10.1016/S0252-9602(12)60202-1.

[25]

G. Molica BisciV. Radulescu and R. Servadei, Competition phenomena for elliptic equations involving a general operator in divergence form, Anal. Appl., 15 (2017), 51-82.  doi: 10.1142/S0219530515500116.

[26]

G. Molica Bisci and D. Repovš, Multiple solutions of $p$-biharmonic equations with Navier boundary conditions, Complex Variables and Elliptic Equations, 59 (2014), 271-284.  doi: 10.1080/17476933.2012.734301.

[27]

T. G. Myers, Thin films with high surface tension, SIAM Review, 40 (1998), 441-462.  doi: 10.1137/S003614459529284X.

[28]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital. Series IX, 3 (2010), 543-582. 

[29]

V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Quantitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601.

[30]

M. R${\rm{\dot{u}}}$žiĄka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer–Verlag Berlin, 2002.

[31]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738.  doi: 10.1016/j.jmaa.2008.07.068.

[32]

A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. T.M.A., 69 (2008), 3629-3636.  doi: 10.1016/j.na.2007.10.001.

[33]

V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66. 

[1]

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

[2]

Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188

[3]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225

[4]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729

[5]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110

[6]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275

[7]

Edcarlos D. Silva, Marcos L. M. Carvalho, Claudiney Goulart. Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1039-1065. doi: 10.3934/dcds.2021146

[8]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170

[9]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[10]

Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615

[11]

Kaiping Liu, Haitao Che, Haibin Chen, Meixia Li. Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022077

[12]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[13]

To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694

[14]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113

[15]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153

[16]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851

[17]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1

[18]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[19]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037

[20]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (516)
  • HTML views (150)
  • Cited by (1)

Other articles
by authors

[Back to Top]