doi: 10.3934/dcdsb.2022109
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Double phase obstacle problems with multivalued convection and mixed boundary value conditions

1. 

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization, and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, China

2. 

Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, China

3. 

Jagiellonian University in Kraków, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348 Kraków, Poland

4. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

5. 

Department of Mathematics, University of Craiova, Street A.I. Cuza 13, 200585 Craiova, Romania

6. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

*Corresponding author: Vicenţiu D. Rădulescu

Received  February 2022 Revised  April 2022 Early access June 2022

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by
$ \mathcal S $
the solution set of the mixed boundary value problem and by
$ \mathcal S_n $
the solution sets of the approximating problems, we establish the following convergence relation
$ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} $
where
$ w $
-
$ \limsup_{n\to\infty}\mathcal S_n $
and
$ s $
-
$ \limsup_{n\to\infty}\mathcal S_n $
stand for the weak and the strong Kuratowski upper limit of
$ \mathcal S_n $
, respectively.
Citation: Shengda Zeng, Vicenţiu D. Rădulescu, Patrick Winkert. Double phase obstacle problems with multivalued convection and mixed boundary value conditions. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022109
References:
[1]

C. O. AlvesP. Garain and V. D. Rădulescu, High perturbations of quasilinear problems with double criticality, Math. Z., 299 (2021), 1875-1895.  doi: 10.1007/s00209-021-02757-z.

[2]

V. Ambrosio and V. D. Rădulescu, Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. (9), 142 (2020), 101-145.  doi: 10.1016/j.matpur.2020.08.011.

[3]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.

[4]

A. BahrouniV. D. Rădulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.  doi: 10.1088/1361-6544/ab0b03.

[5]

A. BahrouniV. D. Rădulescu and P. Winkert, A critical point theorem for perturbed functionals and low perturbations of differential and nonlocal systems, Adv. Nonlinear Stud., 20 (2020), 663-674.  doi: 10.1515/ans-2020-2095.

[6]

A. Bahrouni, V. D. Rădulescu and P. Winkert, Double phase problems with variable growth and convection for the Baouendi-Grushin operator, Z. Angew. Math. Phys., 71 (2020), 15 pp. doi: 10.1007/s00033-020-01412-7.

[7]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[8]

P. Baroni and M. Colombo amd G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.  doi: 10.1090/spmj/1392.

[9]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 48 pp. doi: 10.1007/s00526-018-1332-z.

[10]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.

[11]

M. CenceljV. D. Rădulescu and D. D. Repovš, Double phase problems with variable growth, Nonlinear Anal., 177 (2018), 270-287.  doi: 10.1016/j.na.2018.03.016.

[12]

F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4), 195 (2016), 1917-1959.  doi: 10.1007/s10231-015-0542-7.

[13]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[14]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[15]

Á. Crespo-BlancoL. GasińskiP. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations, 323 (2022), 182-228.  doi: 10.1016/j.jde.2022.03.029.

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

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C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations, 286 (2021), 455-473.  doi: 10.1016/j.jde.2021.03.036.

[18]

M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textbooks in Mathematics, Birkhäuser/Springer, Cham, 2021. doi: 10.1007/978-3-030-75308-5.

[19]

L. Gasiński and N. S. Papageorgiou, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc. Var., 14 (2021), 613-626.  doi: 10.1515/acv-2019-0040.

[20]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8, Chapman & Hall/CRC, Boca Raton, FL, 2005.

[21]

L. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), 1451-1476.  doi: 10.1016/j.jde.2017.03.021.

[22]

L. Gasiński and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal., 195 (2020), 9 pp. doi: 10.1016/j.na.2019.111739.

[23]

L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations, 268 (2020), 4183-4193.  doi: 10.1016/j.jde.2019.10.022.

[24]

L. Gasiński and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations, 274 (2021), 1037-1066.  doi: 10.1016/j.jde.2020.11.014.

[25]

F. Giannessi and A. A. Khan, Regularization of non-coercive quasi variational inequalities, Control Cybernet., 29 (2000), 91-110. 

[26]

A. Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.

[27]

V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658.  doi: 10.1090/S0002-9939-2010-10594-4.

[28]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969.

[29]

W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations, 265 (2018), 4311-4334.  doi: 10.1016/j.jde.2018.06.006.

[30]

G. Marino and P. Winkert, Existence and uniqueness of elliptic systems with double phase operators and convection terms, J. Math. Anal. Appl., 492 (2020), 13 pp. doi: 10.1016/j.jmaa.2020.124423.

[31]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[32]

G. Mingione and V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 41 pp. doi: 10.1016/j.jmaa.2021.125197.

[33]

N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, New York, 2009. doi: 10.1007/b120946.

[34]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc., 147 (2019), 2899-2910.  doi: 10.1090/proc/14466.

[35]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys., 69 (2018), 21 pp. doi: 10.1007/s00033-018-1001-2.

[36]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc., 52 (2020), 546-560.  doi: 10.1112/blms.12347.

[37]

N. S. PapageorgiouC. Vetro and F. Vetro, Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differential Equations, 268 (2020), 4102-4118.  doi: 10.1016/j.jde.2019.10.026.

[38]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., 23 (2021), 18 pp. doi: 10.1142/S0219199720500066.

[39]

N. S. PapageorgiouC. Vetro and F. Vetro, Nonlinear multivalued Duffing systems, J. Math. Anal. Appl., 468 (2018), 376-390.  doi: 10.1016/j.jmaa.2018.08.024.

[40]

N. S. PapageorgiouC. Vetro and F. Vetro, Relaxation for a class of control systems with unilateral constraints, Acta Appl. Math., 167 (2020), 99-115.  doi: 10.1007/s10440-019-00270-4.

[41]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis. An Introduction, De Gruyter Graduate, De Gruyter, Berlin, 2018.

[42]

K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math., 20 (2018), 14 pp. doi: 10.1142/S0219199717500237.

[43]

V. D. Rădulescu, Isotropic and anistropic double-phase problems: Old and new, Opuscula Math., 39 (2019), 259-279.  doi: 10.7494/OpMath.2019.39.2.259.

[44]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987.

[45]

C. Vetro, Parametric and nonparametric $A$-Laplace problems: Existence of solutions and asymptotic analysis, Asymptot. Anal., 122 (2021), 105-118.  doi: 10.3233/ASY-201612.

[46]

C. Vetro and F. Vetro, On problems driven by the $(p(\cdot), q(\cdot))$-Laplace operator, Mediterr. J. Math., 17 (2020), 11 pp. doi: 10.1007/s00009-019-1448-1.

[47]

S. ZengY. BaiL. Gasiński and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal., 10 (2021), 659-672.  doi: 10.1515/anona-2020-0155.

[48]

S. Zeng, Y. Bai, L. Gasiński and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations, 59 (2020), 18 pp. doi: 10.1007/s00526-020-01841-2.

[49]

S. Zeng, L. Gasiński, P. Winkert and Y. Bai, Existence of solutions for double phase obstacle problems with multivalued convection term, J. Math. Anal. Appl., 501 (2021), 12 pp. doi: 10.1016/j.jmaa.2020.123997.

[50]

Q. Zhang and V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), 118 (2018), 159-203.  doi: 10.1016/j.matpur.2018.06.015.

[51]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[52]

V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. (N.Y.), 173 (2011), 463-570.  doi: 10.1007/s10958-011-0260-7.

show all references

References:
[1]

C. O. AlvesP. Garain and V. D. Rădulescu, High perturbations of quasilinear problems with double criticality, Math. Z., 299 (2021), 1875-1895.  doi: 10.1007/s00209-021-02757-z.

[2]

V. Ambrosio and V. D. Rădulescu, Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. (9), 142 (2020), 101-145.  doi: 10.1016/j.matpur.2020.08.011.

[3]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.

[4]

A. BahrouniV. D. Rădulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.  doi: 10.1088/1361-6544/ab0b03.

[5]

A. BahrouniV. D. Rădulescu and P. Winkert, A critical point theorem for perturbed functionals and low perturbations of differential and nonlocal systems, Adv. Nonlinear Stud., 20 (2020), 663-674.  doi: 10.1515/ans-2020-2095.

[6]

A. Bahrouni, V. D. Rădulescu and P. Winkert, Double phase problems with variable growth and convection for the Baouendi-Grushin operator, Z. Angew. Math. Phys., 71 (2020), 15 pp. doi: 10.1007/s00033-020-01412-7.

[7]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[8]

P. Baroni and M. Colombo amd G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.  doi: 10.1090/spmj/1392.

[9]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 48 pp. doi: 10.1007/s00526-018-1332-z.

[10]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.

[11]

M. CenceljV. D. Rădulescu and D. D. Repovš, Double phase problems with variable growth, Nonlinear Anal., 177 (2018), 270-287.  doi: 10.1016/j.na.2018.03.016.

[12]

F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4), 195 (2016), 1917-1959.  doi: 10.1007/s10231-015-0542-7.

[13]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[14]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[15]

Á. Crespo-BlancoL. GasińskiP. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations, 323 (2022), 182-228.  doi: 10.1016/j.jde.2022.03.029.

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

[17]

C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations, 286 (2021), 455-473.  doi: 10.1016/j.jde.2021.03.036.

[18]

M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textbooks in Mathematics, Birkhäuser/Springer, Cham, 2021. doi: 10.1007/978-3-030-75308-5.

[19]

L. Gasiński and N. S. Papageorgiou, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc. Var., 14 (2021), 613-626.  doi: 10.1515/acv-2019-0040.

[20]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8, Chapman & Hall/CRC, Boca Raton, FL, 2005.

[21]

L. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), 1451-1476.  doi: 10.1016/j.jde.2017.03.021.

[22]

L. Gasiński and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal., 195 (2020), 9 pp. doi: 10.1016/j.na.2019.111739.

[23]

L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations, 268 (2020), 4183-4193.  doi: 10.1016/j.jde.2019.10.022.

[24]

L. Gasiński and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations, 274 (2021), 1037-1066.  doi: 10.1016/j.jde.2020.11.014.

[25]

F. Giannessi and A. A. Khan, Regularization of non-coercive quasi variational inequalities, Control Cybernet., 29 (2000), 91-110. 

[26]

A. Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.

[27]

V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658.  doi: 10.1090/S0002-9939-2010-10594-4.

[28]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969.

[29]

W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations, 265 (2018), 4311-4334.  doi: 10.1016/j.jde.2018.06.006.

[30]

G. Marino and P. Winkert, Existence and uniqueness of elliptic systems with double phase operators and convection terms, J. Math. Anal. Appl., 492 (2020), 13 pp. doi: 10.1016/j.jmaa.2020.124423.

[31]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[32]

G. Mingione and V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 41 pp. doi: 10.1016/j.jmaa.2021.125197.

[33]

N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, New York, 2009. doi: 10.1007/b120946.

[34]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc., 147 (2019), 2899-2910.  doi: 10.1090/proc/14466.

[35]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys., 69 (2018), 21 pp. doi: 10.1007/s00033-018-1001-2.

[36]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc., 52 (2020), 546-560.  doi: 10.1112/blms.12347.

[37]

N. S. PapageorgiouC. Vetro and F. Vetro, Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differential Equations, 268 (2020), 4102-4118.  doi: 10.1016/j.jde.2019.10.026.

[38]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., 23 (2021), 18 pp. doi: 10.1142/S0219199720500066.

[39]

N. S. PapageorgiouC. Vetro and F. Vetro, Nonlinear multivalued Duffing systems, J. Math. Anal. Appl., 468 (2018), 376-390.  doi: 10.1016/j.jmaa.2018.08.024.

[40]

N. S. PapageorgiouC. Vetro and F. Vetro, Relaxation for a class of control systems with unilateral constraints, Acta Appl. Math., 167 (2020), 99-115.  doi: 10.1007/s10440-019-00270-4.

[41]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis. An Introduction, De Gruyter Graduate, De Gruyter, Berlin, 2018.

[42]

K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math., 20 (2018), 14 pp. doi: 10.1142/S0219199717500237.

[43]

V. D. Rădulescu, Isotropic and anistropic double-phase problems: Old and new, Opuscula Math., 39 (2019), 259-279.  doi: 10.7494/OpMath.2019.39.2.259.

[44]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987.

[45]

C. Vetro, Parametric and nonparametric $A$-Laplace problems: Existence of solutions and asymptotic analysis, Asymptot. Anal., 122 (2021), 105-118.  doi: 10.3233/ASY-201612.

[46]

C. Vetro and F. Vetro, On problems driven by the $(p(\cdot), q(\cdot))$-Laplace operator, Mediterr. J. Math., 17 (2020), 11 pp. doi: 10.1007/s00009-019-1448-1.

[47]

S. ZengY. BaiL. Gasiński and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal., 10 (2021), 659-672.  doi: 10.1515/anona-2020-0155.

[48]

S. Zeng, Y. Bai, L. Gasiński and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations, 59 (2020), 18 pp. doi: 10.1007/s00526-020-01841-2.

[49]

S. Zeng, L. Gasiński, P. Winkert and Y. Bai, Existence of solutions for double phase obstacle problems with multivalued convection term, J. Math. Anal. Appl., 501 (2021), 12 pp. doi: 10.1016/j.jmaa.2020.123997.

[50]

Q. Zhang and V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), 118 (2018), 159-203.  doi: 10.1016/j.matpur.2018.06.015.

[51]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[52]

V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. (N.Y.), 173 (2011), 463-570.  doi: 10.1007/s10958-011-0260-7.

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