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

  • * Corresponding author: J. Yin

    * Corresponding author: J. Yin 
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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  • 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.

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


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