Variable exponents anisotropic nonlinear elliptic systems with -data

Abstract

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary ${\mathring{W}}^{1,\overrightarrow{p}(\cdot )}(\mathrm{\Omega },{\mathbb{R}}^m)$ for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with $L^{\overrightarrow{p^{\prime }}(\cdot )}$ data for the datum f (i.e. $\in L^{\overrightarrow{p^{\prime }}(\cdot )}(\mathrm{\Omega },{\mathbb{R}}^m)$), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives ${\mathrm{\partial }}_{i}u,i=1,\dots ,N$.

Keywords:

• Variable exponents
• nonlinear elliptic systems
• anisotropic Lebesgue–Sobolev spaces
• distributional solution
• $L^{\overrightarrow{p^{\prime }}(\cdot )}$-data
• existence

2020 Mathematics Subject Classifications:

• 35J60
• 35J66
• 35J67
• 47H10

Acknowledgments

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

Disclosure statement

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