Existence of solutions for a class of degenerate elliptic equations in $P(x)$-Sobolev spaces

Benali Aharrouch, Mohamed Boukhrij, Jaouad Bennouna


We study the Dirichlet problem for degenerate elliptic equations of the form \begin{equation*} - \mbox{div} a(x,u,\nabla u)+ H(x,u,\nabla u)= f \quad \mbox{in } \Omega, \end{equation*} where $ a(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown $u$, and $H(x,u,\nabla u)$ is a nonlinear term without sign condition. Under suitable conditions on $a$ and $H$, we prove the existence of bounded and unbounded solution for a datum $f\in L^m$, with $1\leq m\leq \infty$.


Weak and entropy solutions; degenerate elliptic equations; Sobolev spaces with variable exponent; Stampacchia methods

Full Text:



L. Aharouch, A. Benkirane and M. Rhoudaf, Existence results for some unilateral problems without sign condition with obstacle free in Orlicz spaces, Nonlinear Anal. 68 (2008), 2362–380.

L. Aharouch and J. Bennouna, Existence and uniqueness of solutions of unilateral problems in Orlicz spaces, Nonlinear Anal. 72 (2010), no. 9–10, 3553–3565.

E. Azroul, H. Hjiaj and A. Touzani, Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations, Electron. J. Differential Equations 68 (2013), 1–27.

M.B. Benboubker, E. Azroul and A. Barbara, Quasilinear elliptic problems with nonstandard growth, Electron. J. Differential Equations 62 (2011), 1–16.

M. Bendahmane and F. Mokhtari, Nonlinear elliptic systems with variable exponents and measure data, Moroccan J. Pure Appl. Anal. (MJPAA) 1 (2015), 108–125.

M. Bendahmane and P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data, Nonlinear Anal. 70 (2009), 567–583.

Ph. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J.L. Vazquez, An L1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1995), 241–273.

L. Boccardo, T. Gallouet and L.Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73, 203–223 (1997).

L. Boccardo, F. Murat and J.P. Puel, Existence de solutions non bornes pour certaines equations quasi-lineaires, Portugal. Math. 41 (1982), 507–534.

L. Boccardo, S. Segura and C. Trombeti, Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term, J. Math. Pures Appl. 80 (2001), 919–940.

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Heidelberg, 2011.

X.L. Fan and D. Zhao, On the generalised Orlicz–Sobolev space W k.p( · ) (Ω), J. Gansu Educ. College 12 (1998), 1–6.

O. Kováčik and J. Rákosnı́k, On spaces Lp( · ) and W 1.p( · ) , Czechoslovak Math. J. 41 (1991), no. 116, 592–618.

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

I. Nyanquini and S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afr. Mat. 23 (2012), no. 2, 205–228.

A. Porretta, Nonlinear equations with natural growth terms and measure data, Electron.J. Differ. Equ. Conf. 9 (2002), 183–202.

A. Porretta and S. Segura de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl. (9) 85 (2006), no. 3, 465–492.

J.M. Rakotoson, Réarrangement relatif dans les équations qausi-linéaires avec un second membre distribution: application à un théorème d’existence et de régularité, J. Differential Equations 66 (1987), 391–419.

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258.

D. Zhao and X.L. Fan, On the Nemytskiı̆ operators from Lp1 ( · ) to Lp2 ( · ) , J. Lanzhou Univ. 34 (1998), 1–5.

D. Zhao, W.J. Qiang and X.L. Fan, On generalized Orlicz spaces Lp ( · )(Ω), J. Gansu Sci. 9 (1997), 1–7.

W. Zou and F. Li, Existence of solutions for degenerate quasilinear elliptic equations, J. Nonlinear Anal. 73 (2010), 3069–3082.


  • There are currently no refbacks.

Partnerzy platformy czasopism