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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 & 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.   Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.   Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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