Anisotropic elliptic system with variable exponents and degenerate coercivity with L^m data

Abstract

In a bounded open domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, where $N\ge 2$, with Lipschitz boundary $\mathrm{\partial \Omega }$, we consider the Dirichlet problem for the elliptic system given by $\{\begin{array}{ll}{\displaystyle -\sum _{i=1}^N{D}_i(a_i(x,u(x),D_{i}u(x)))+F(x,u)=f(x),}& x\in \mathrm{\Omega },\\ u(x)=0,& x\in \mathrm{\partial \Omega },\end{array}$here, $u:\mathrm{\Omega }\to {\mathbb{R}}^n$, $n\ge 2$, represents a vector-valued function, $D_{i}u=\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}$ denotes the partial derivative of u with respect to $x_i$, and the vector fields $a_i:\mathrm{\Omega }\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $F:\mathrm{\Omega }\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and $L^m$ data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to $L^m(\mathrm{\Omega };{\mathbb{R}}^n)$ with 1<m<N. To analyze this problem, we work with an appropriate functional setting that involves anisotropic Sobolev spaces with variable exponents and weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

Keywords:

• Elliptic system
• degenerate equations
• nonlinear elliptic equations
• anisotropic equations
• $L^m$ data

COMMUNICATED BY:

• M. A. Ragusa

AMS Classifications:

• 35J50
• 35J70
• 35J60

Acknowledgments

The authors would like to thank the referees for their comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).