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December  2020, 9(4): 1057-1071. doi: 10.3934/eect.2020056

Differential inclusion problems with convolution and discontinuous nonlinearities

Yongjian Liu 1, , Zhenhai Liu 2, and  Dumitru Motreanu 3,1,,

1. 

Department of Mathematics, Yulin Normal University, Yulin, Guangxi, 537000, China
2. 

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, China
3. 

Department of Mathematics, Université de Perpignan, 52 Avenue Paul Alduy, 66860, Perpignan, France

* Corresponding author

Received  October 2019 Revised  January 2020 Published  December 2020 Early access  May 2020

The paper investigates a new type of differential inclusion problem driven by a weighted (p, q)-Laplacian and subject to Dirichlet boundary condition. The problem fully depends on the solution and its gradient. The main novelty is that the problem exhibits simultaneously a nonlocal term involving convolution with the solution and a multivalued term describing discontinuous nonlinearities for the solution. Results stating existence, uniqueness and dependence on parameters are established.

Keywords: Differential inclusion, hemivariational inequality, quasilinear elliptic equation, weighted (p, q)-Laplacian, convolution, discontinuous nonlinearity.
Mathematics Subject Classification: Primary: 35J92, 35J87; Secondary: 49J52.
Citation: Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations and Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056
References:
[1]

D. Averna, D. Motreanu and E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.  doi: 10.1016/j.aml.2016.05.009.

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[3]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[4]

K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0.

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.

[6]

Z. Liu and D. Motreanu, Inclusion problems via subsolution-supersolution method with applications to hemivariational inequalities, Appl. Anal., 97 (2018), 1454-1465.  doi: 10.1080/00036811.2017.1408076.

[7]

S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition, Nonlinear Anal., 187 (2019), 159-169.  doi: 10.1016/j.na.2019.04.008.

[8]

D. Motreanu and V. V. Motreanu, Non-variational elliptic equations involving $(p, q)$-Laplacian, convection and convolution, Pure Appl. Funct. Anal., preprint.

[9]

D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, in Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9.

[10]

D. Motreanu and Z. Peng, Doubly coupled systems of parabolic hemivariational inequalities: Existence and extremal solutions, Nonlinear Anal., 181 (2019), 101-118.  doi: 10.1016/j.na.2018.11.005.

[11]

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, in Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.

[12]

M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[13] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. 
[14]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

D. Averna, D. Motreanu and E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.  doi: 10.1016/j.aml.2016.05.009.

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[3]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[4]

K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0.

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.

[6]

Z. Liu and D. Motreanu, Inclusion problems via subsolution-supersolution method with applications to hemivariational inequalities, Appl. Anal., 97 (2018), 1454-1465.  doi: 10.1080/00036811.2017.1408076.

[7]

S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition, Nonlinear Anal., 187 (2019), 159-169.  doi: 10.1016/j.na.2019.04.008.

[8]

D. Motreanu and V. V. Motreanu, Non-variational elliptic equations involving $(p, q)$-Laplacian, convection and convolution, Pure Appl. Funct. Anal., preprint.

[9]

D. Motreanu and P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, in Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9.

[10]

D. Motreanu and Z. Peng, Doubly coupled systems of parabolic hemivariational inequalities: Existence and extremal solutions, Nonlinear Anal., 181 (2019), 101-118.  doi: 10.1016/j.na.2018.11.005.

[11]

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, in Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.

[12]

M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[13] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. 
[14]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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