African Diaspora Journal of Mathematics

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

S. Ouaro and A. Tchousso

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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


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