Weak solutions for anisotropic nonlinear elliptic equations with variable exponents

Abstract

We study the anisotropic boundary-value problem $$displaylines{ -sum^{N}_{i=1}frac{partial}{partial x_{i}}a_{i}(x,frac{partial}{partial x_{i}}u)=f quad hbox{in } Omega, cr u=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded domain in $mathbb{R}^{N}$ $(Ngeq 3)$. We obtain the existence and uniqueness of a weak energy solution for $fin L^{infty}(Omega)$, and the existence of weak energy solution for general data $f$ dependent on $u$.

Keywords

• Anisotropic Sobolev spaces
• weak energy solution
• variable exponents
• electrorheological fluids