Positive weak solutions of uniformly elliptic inequalities and applications to population models

Abstract

By using the uniformly ellipticity of linear partial differential operators (which contain p-Laplacian operators with $1<p<\mathrm{\infty }$), weak comparison principle and Urysohn's lemma, in the paper, we introduce and study a class of new variational inequalities involving uniformly elliptic operators and demicontinuous S-contractive operators to extend some corresponding tasks and scope on population genetics. Further, as applications, we employ the main results presented in this paper to second-order uniformly elliptic inequalities and a population model of one species arising in disrupted environments.

Keywords:

• Solvability-approximation
• weak comparison principle
• uniformly elliptic inequality
• demicontinuous S-contractive operator
• nonzero positive weak solution

AMS Subject Classifications:

• 49J40
• 35R45
• 47J20
• 92D25

Acknowledgements

We thank Prof. Kunquan Lan for careful guidance, and the referees’ and Editor's valuable comments and suggestions to improve our paper.

Disclosure statement

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

Additional information

Funding

This work was supported by the Sichuan Science and Technology Program [grant number 2019YJ0541], which comes from Science & Technology Department of Sichuan Province, Sichuan, PR China.