## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 11, Number 2 (2011), 36-64.

### Well-Posedness Result For a Nonlinear Elliptic Problem Involving Variable Exponent and Robin Type Boundary Condition

S. Ouaro and A. Tchousso

#### Abstract

In this work we study the following nonlinear elliptic boundary value problem, $b(u)-div \; a(x,\nabla u)=f \hbox{ in }\Omega$, $a(x,\nabla u).\eta=-\left|u\right|^{p(x)-2}u \hbox{ on }\partial \Omega$, where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 1$ with smooth boundary $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$, the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2 (2011), 36-64.

**Dates**

First available in Project Euclid: 6 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1323180285

**Mathematical Reviews number (MathSciNet)**

MR2862564

**Zentralblatt MATH identifier**

1242.35110

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations

Secondary: 35D30: Weak solutions 35B38 35J60

**Keywords**

Lebesgue and Sobolev spaces with variable exponent Weak solution Entropy solution Robin type boundary condition

#### Citation

Ouaro, S.; Tchousso, A. Well-Posedness Result For a Nonlinear Elliptic Problem Involving Variable Exponent and Robin Type Boundary Condition. Afr. Diaspora J. Math. (N.S.) 11 (2011), no. 2, 36--64. https://projecteuclid.org/euclid.adjm/1323180285