Entropy solution for double phase elliptic (p, q)-Laplacien problem with Right-Hand Side Measure
Abstract
In this work, we establish the existence of entropy solutions for a double phase elliptic (p,q)-
Laplacian problem, subject to Dirichlet boundary conditions and measure data on the right-hand side. Our
main strategy combines the variational method with the framework of Sobolev spaces, by verifying the conditions of the Minty–Browder theorem.
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References
1. A.sabri and A.Jamea. Weak solution for nonlinear degenerate elliptic problem with dirichlet-type boundary condition
in weighted sobolev spaces. Mathematica Bohemica, 26(3):4-18, 2021. https://doi.org/10.21136/MB.2021.0004-20.
2. G.Dal Maso, F.Murat, L.Orsina and A.Prignet. Renormalized solutions of elliptic equations with general measure
data.Ann. Scuola Norm. Sup. Pisa Cl. Sci. ol. XXVIII (1999), pp. 741-808.
3. L. Boccardo, T. Gallouet, L.Orsina:Existence and uniqueness of entropy solutions for nonlinear elliptic equations with
measure data; Ann. Inst. Henri Poincar´e, Vol. 13, nO 5, 1996, p. 539-551, 1992.
4. L. Boccardo, T. Gallouet F. Murat : Unicite de la solution de certaines ´equations. elliptiques non lineaires; C. R. Acad.
Sci. Paris, t. 315, S´erie I, p. 1159-1164,(1992).
5. A. C. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Paran. Mat. 26, 117:132(2008).
6. P. A. Hasto; The p(x)-Laplacian and applications, Proceedings of the International Conference on Geometric Function
Theory. 15, 53:62(2007).
7. M. Ruzicka: Electrorheological Fluids, Modeling and Mathematical Theory, Lectures Notes in Math.,Vol. 1748, Springer,
Berlin, 2000.
8. Ph. B´enilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez: An L1
- theory of existence and uniqueness
of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 22 (2) (1995)
241273.
9. A. Sabri, A. Jamea, H. A. Talibi, Weak solution for nonlinear degenerate elliptic problem with Dirichlet-type boundary
condition in weighted Sobolev spaces, Mathematica Bohemica, Vol. 147, No. 1, pp. 113-129, 202
10. A. Sabri, A. Jamea, Rothe time-discretization method for a nonlinear parabolic p(u) -Laplacian problem with Fouriertype boundary condition and L1
-data. Ricerche mat (2020). https://doi.org/10.1007/s11587-020-00544-2 (6) (2010)
13761400.
11. C. Zhang: Entropy solutions for nonlinear elliptic equations with variable exponents. Electronic Journal of Differential
Equations 2014 (92) (2014) 1-14.
12. X. Fan, D. Zhao: On the spaces Lp(x)
(W) and W m,p(x)
(W). Journal of Mathematical Analysis and Applications 263
(2) (2001) 424-446.
13. M. Sanch´on, J.M. Urbano: Entropy solutions for the p(x)-Laplace equation. Transactions of the American Mathematical
Society 361 (12) (2009) 6387-6405.
14. V. Mustonen P. Drabek, A. Kufner. Pseudo-monotonicity and degenerated or singular elliptic operators, bull. austral.
Math. Soc, (58):213-221, 1998
15. G. Stampacchia, Le probl‘eme de Dirichlet pour les ´equations elliptiques du second ordre ‘a coefficients discontinus.
Ann. Inst. Fourier 15 (1965), 189–258
16. H. Brezis and W. Strauss, Semilinear elliptic equations in L1
. J. Math. Soc. Japan 25 (1973), 565–590.
17. H. Brezis, Some variational problems of the Thomas-Fermi type. In Variational Inequalities (Ed. Cottle, Gianessi-Lions)
(Wiley, New York) (1980), 53–73.
18. P. B´enilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation. Journal of Evolution Equations
3 n° 4 (2003), 673–770.
in weighted sobolev spaces. Mathematica Bohemica, 26(3):4-18, 2021. https://doi.org/10.21136/MB.2021.0004-20.
2. G.Dal Maso, F.Murat, L.Orsina and A.Prignet. Renormalized solutions of elliptic equations with general measure
data.Ann. Scuola Norm. Sup. Pisa Cl. Sci. ol. XXVIII (1999), pp. 741-808.
3. L. Boccardo, T. Gallouet, L.Orsina:Existence and uniqueness of entropy solutions for nonlinear elliptic equations with
measure data; Ann. Inst. Henri Poincar´e, Vol. 13, nO 5, 1996, p. 539-551, 1992.
4. L. Boccardo, T. Gallouet F. Murat : Unicite de la solution de certaines ´equations. elliptiques non lineaires; C. R. Acad.
Sci. Paris, t. 315, S´erie I, p. 1159-1164,(1992).
5. A. C. Cavalheiro, Weighted Sobolev spaces and degenerate elliptic equations, Bol. Soc. Paran. Mat. 26, 117:132(2008).
6. P. A. Hasto; The p(x)-Laplacian and applications, Proceedings of the International Conference on Geometric Function
Theory. 15, 53:62(2007).
7. M. Ruzicka: Electrorheological Fluids, Modeling and Mathematical Theory, Lectures Notes in Math.,Vol. 1748, Springer,
Berlin, 2000.
8. Ph. B´enilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez: An L1
- theory of existence and uniqueness
of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 22 (2) (1995)
241273.
9. A. Sabri, A. Jamea, H. A. Talibi, Weak solution for nonlinear degenerate elliptic problem with Dirichlet-type boundary
condition in weighted Sobolev spaces, Mathematica Bohemica, Vol. 147, No. 1, pp. 113-129, 202
10. A. Sabri, A. Jamea, Rothe time-discretization method for a nonlinear parabolic p(u) -Laplacian problem with Fouriertype boundary condition and L1
-data. Ricerche mat (2020). https://doi.org/10.1007/s11587-020-00544-2 (6) (2010)
13761400.
11. C. Zhang: Entropy solutions for nonlinear elliptic equations with variable exponents. Electronic Journal of Differential
Equations 2014 (92) (2014) 1-14.
12. X. Fan, D. Zhao: On the spaces Lp(x)
(W) and W m,p(x)
(W). Journal of Mathematical Analysis and Applications 263
(2) (2001) 424-446.
13. M. Sanch´on, J.M. Urbano: Entropy solutions for the p(x)-Laplace equation. Transactions of the American Mathematical
Society 361 (12) (2009) 6387-6405.
14. V. Mustonen P. Drabek, A. Kufner. Pseudo-monotonicity and degenerated or singular elliptic operators, bull. austral.
Math. Soc, (58):213-221, 1998
15. G. Stampacchia, Le probl‘eme de Dirichlet pour les ´equations elliptiques du second ordre ‘a coefficients discontinus.
Ann. Inst. Fourier 15 (1965), 189–258
16. H. Brezis and W. Strauss, Semilinear elliptic equations in L1
. J. Math. Soc. Japan 25 (1973), 565–590.
17. H. Brezis, Some variational problems of the Thomas-Fermi type. In Variational Inequalities (Ed. Cottle, Gianessi-Lions)
(Wiley, New York) (1980), 53–73.
18. P. B´enilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation. Journal of Evolution Equations
3 n° 4 (2003), 673–770.
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