Anisotropic variable exponent Sobolev spaces and -Laplacian equations

Abstract

In this article, some basic properties of anisotropic variable exponent Sobolev spaces and -Laplacian operators are presented. An application of Ricceri's variational principle to -Laplacian equations is given and an eigenvalue problem involving the -Laplacian is studied.

Keywords:

• anisotropic
• variable exponent
• Sobolev space
• embedding
• elliptic equation

AMS Subject Classifications::

• 46E35
• 46E30
• 35J70
• 58E30

Acknowledgements

The author visited Vladimir State Pedagogical University in Russia as a visiting researcher from October 1991 till April 1992. During his visiting, the author started research on variable exponent problems under Prof V.V. Zhikov's supervision. He would like to express his gratitude to Prof V.V. Zhikov. The research was supported by National Natural Science Foundation of China (No. 10971087).