Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources

Shanming Ji; Jingxue Yin; Yutian Li

doi:10.3934/dcds.2018100

Article Contents

2018, Volume 38, Issue 5: 2411-2439. Doi: 10.3934/dcds.2018100

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Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources

1.
School of Mathematics, South China University of Technology, Guangzhou 510640, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3.
School of Science and Engineering, Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China

* Corresponding author: J. Yin
* Corresponding author: J. Yin

Received: July 31, 2017

Revised: December 31, 2017

Published: June 2018

The first author is supported by the Fundamental Research Funds for the Central Universities (No. 2017BQ109), China Postdoctoral Science Foundation (No. 2017M610517) and NSFC Grant No. 11701184. The second author is supported by NSFC Grant No. 11771156

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Abstract

In this paper we study the existence of time periodic solutions for the evolutionary weighted $p$-Laplacian with a nonlinear periodic source in a bounded domain containing the origin. We show that there is a critical exponent $q_c = q_c(α,β) = \frac{(N+β)p}{N+α-p}-1$ and a singular exponent $q_s = p-1$: there exists a positive periodic solution when $0<q<q_c$ and $q\ne q_s$; while there is no positive periodic solution when $q≥ q_c$. The case when $q = q_s$ is completely different from the remaining case $q\ne q_s$, the problem may or may not have solutions depending on the coefficients of the equation.

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Mathematics Subject Classification: Primary: 35B10, 35K10; Secondary: 35K65.

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