Capacity solution for an elliptic coupled system with lower term in Orlicz spaces

Yassine Ahakkoud; Jaouad Bennouna; Mhamed El Massoudi

doi:10.24193/subbmath.2025.2.07

Authors

Yassine Ahakkoud University Sidi Mohamed Ben Abdellah, Fez, Morocco
Jaouad Bennouna University Sidi Mohamed Ben Abdellah, Fez, Morocco
Mhamed El Massoudi University Sidi Mohamed Ben Abdellah, Fez, Morocco

DOI:

https://doi.org/10.24193/subbmath.2025.2.07

Keywords:

Perturbed coupled system, Capacity solution, Nonlinear elliptic equations, Weak solution, Orlicz-Soblev spaces

Abstract

In this paper, we will deal with the capacity solution for a nonlinear elliptic coupled system with a Leray-Lions operator \(Au=-\mbox{div }\sigma(x,u,\nabla u)\) acting from Orlicz-Sobolev spaces \(W_{0}^{1}L_{M}(\Omega)\) into its dual, where \(M\) is an \(N\)-function.

References

[1] Aberqi, A., Bennouna, J., Elmassoudi, M., Sub-supersolution method for nonlinear elliptic equation with non-coercivity in divergentiel form in Orlicz spaces, AIP Conference Proceedings 2074, 020004 (2019).

[2] Aberqi, A., Bennouna, J., Elmassoudi, M., Existence of entropy solutions in Musielak- Orlicz spaces via a sequence of penalized equations, Bol. Soc. Parana. Mat. (3s.), 38(6)(2020), 203-238.

[3] Aberqi, A., Bennouna, J., Elmassoudi, M., Hammoumi, M., Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces, Monatsh. Math., 189(2019), 195-219.

[4] Adams, R.A., Sobolev Spaces, New York (NY), Academic Press, 1975.

[5] Aharouch, L., Bennouna, J., Existence and uniqueness of solutions of unilateral problems in Orlicz spaces, Nonlinear Anal. TMA, 72(2010), 3553-3565.

[6] Azroul, E., Redwane, H., Rhoudaf, M., Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces, Port. Math. (N.S.), 66(2009), no. 1, 29-63.

[7] Bahari, M., El Arabi, R., Rhoudaf, M., Capacity solution for a perturbed nonlinear coupled system, Ric. Mat., 69(1)(2020), 215-233.

[8] Bass, J., Thermoelectricity, In: Parer, S.P. (ed.) Mc Graw-Hill Encyclopedia of Physics, McGraw-Hill, New York, 1982.

[9] Benkirane, A., Elmahi, A., Almost every where convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Anal. TMA, 11(28)(1997), 1769-1784.

[10] Gonz alez Montesinos, M.T., Ortegon Gallego, F., Existence of a capacity solution to a coupled nonlinear parabolic-elliptic system, Commun. Pure Appl. Anal., 6(1)(2007), 23-42.

[11] Gossez, J.-P., Mustonen, V., Variational inequalities in Orlics-spaces, Nonlinear Anal., 11(1987), 379-492.

[12] Moussa, H., Orteg on Gallego, F., Rhoudaf, M. , Capacity solution to a nonlinear elliptic coupled system in Orlicz-Sobolev spaces, Mediterr. J. Math., 17(67)(2020), 1-28.

[13] Xu, X., A strongly degenrate system involing an equation of parabolic type and an equation of elliptic type, Comm. Partial Differential Equations, 18(1993), 1-21.

[14] Xu, X., On the existence of bounded temperature in the thermistor problem with degeneracy, Nonlinear Anal. TMA, 42(2000), 199-213.

[15] Young, J.M., Steady state Joule heating with temperature dependent conductivities, Applied scientific Research, 42(1986), 55-65.