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Uniqueness of positive radial solutions for a class of infinite semipositone $ p$-Laplacian problems in a ball


Authors: K. D. Chu, D. D. Hai and R. Shivaji
Journal: Proc. Amer. Math. Soc. 148 (2020), 2059-2067
MSC (2010): Primary 34B16; Secondary 34B18, 35J62
DOI: https://doi.org/10.1090/proc/14886
Published electronically: January 13, 2020
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Abstract: We prove uniqueness of positive radial solutions to the $ p$-Laplacian problem

$\displaystyle \left \{ \begin {array}{c} -\Delta _{p}u=\lambda f(u)\text { in }\Omega , \\ u=0\text { on }\partial \Omega , \end{array} \right .$    

where $ \Delta _{p}u=\operatorname {div}(\vert\nabla u\vert^{p-2}\nabla u),p\geq 2,\ \Omega $ is the open unit ball in $ R^{N}, N>1,\ f:(0,\infty )\rightarrow \mathbb{R}$ is concave, $ p$-sublinear at $ \infty $ with infinite semipositone structure at 0, and $ \lambda $ is a large parameter.

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Additional Information

K. D. Chu
Affiliation: Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Email: chuduckhanh@tdtu.edu.vn

D. D. Hai
Affiliation: Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762
Email: dang@math.msstate.edu

R. Shivaji
Affiliation: Department of Mathematics and Statisitics, University of North Carolina at Greensboro, Greensboro, North Carolina 27402
Email: shivaji@uncg.edu

DOI: https://doi.org/10.1090/proc/14886
Keywords: Uniqueness, infinite semipositone, \(p\)-Laplacian, positive solutions
Received by editor(s): June 25, 2019
Received by editor(s) in revised form: September 12, 2019
Published electronically: January 13, 2020
Communicated by: Wenxian Shen
Article copyright: © Copyright 2020 American Mathematical Society