## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 13, Number 2 (2012), 23-44.

### Entropy Solution for Some $p(x)$-Quasilinear Problem with Right-Hand Side Measure

E. Azroul, M. B. Benboubker, and M. Rhoudaf

#### Abstract

In this paper we study the existence of entropy solution for the following $p(x)$-quasilinear elliptic problem $$ \mbox{div}(a(x,u,\nabla u))+ g(x,u,\nabla u) = \mu$$ where the right-hand side $\mu$ is a measure, which admits a decomposition in $L^{1}(\Omega)+W^{-1,p'(x)}(\Omega)$ and $g(x,s,\xi)$ is a nonlinear term which has a growth condition with respect to $\xi$ and has no growth with respect to $s$ while satisfying a sign condition on $s$.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2 (2012), 23-44.

**Dates**

First available in Project Euclid: 2 November 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1351864731

**Mathematical Reviews number (MathSciNet)**

MR3006751

**Zentralblatt MATH identifier**

1272.35100

**Subjects**

Primary: 35J15: Second-order elliptic equations

Secondary: 35J20: Variational methods for second-order elliptic equations 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

**Keywords**

Quasilinear elliptic equation Sobolev spaces with variable exponent entropy solution, truncations

#### Citation

Azroul, E.; Benboubker, M. B.; Rhoudaf, M. Entropy Solution for Some $p(x)$-Quasilinear Problem with Right-Hand Side Measure. Afr. Diaspora J. Math. (N.S.) 13 (2012), no. 2, 23--44. https://projecteuclid.org/euclid.adjm/1351864731