Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

Trad Alotaibi; D. D. Hai; R. Shivaji

doi:10.3934/cpaa.2020131

Article Contents

2020, Volume 19, Issue 9: 4655-4666. Doi: 10.3934/cpaa.2020131

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$ \mathbb{R}^N $

Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

1.
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
2.
Department of Mathematics and Statisitics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA

^* Corresponding author
^* Corresponding author

Received: February 28, 2019

Revised: October 31, 2019

Published: June 2020

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Abstract

We prove the existence of positive radial solutions to the problem

$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $

where$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $ $ f:(0,\infty )\rightarrow \mathbb{R} $ is $ p $-superlinear at $ \infty $ with possible singularity at $ 0, $ and $ \lambda $ is a small positive parameter. A nonexistence result is also established when $ f $ has semipositone structure at $ 0. $

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Mathematics Subject Classification: Primary: 35J66, 35J92; Secondary: 35J75.

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