Solvable optimization problems involving a p-Laplacian type operator

Abstract

This paper is concerned with maximization and minimization problems related to a boundary value problem involving a p-Laplacian type operator. These optimization problems are formulated relative to the rearrangement of a fixed function. Firstly, by introducing a truncated function, we establish the existence and uniqueness of the solution of the boundary value problem involving a p-Laplacian type operator, and then, we show that both optimization problems are solvable under some suitable assumptions. Furthermore, we show that the solution of the minimization problem is unique and has some symmetric property if the domain considered is a ball.

Keywords:

• Optimization
• rearrangement
• symmetric
• p-Laplacian

2010 Mathematics Subject Classifications:

• 35J20
• 35J88
• 49J40

Acknowledgments

The authors express their thanks very much for the valuable comments that are given by the referees.

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 11771319, 11971339], Natural Science Foundation of Jiangsu Province [grant numbers BK20170590, BK20150281], The Natural Science Foundation of the Jiangsu Higher Education Institutions of China [grant number 16KJB110020] and Jiangsu Provincial Government Scholarship for Studying Abroad and the Research Grants Council of Hong Kong [grant number PolyU 152165/18E].