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

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  • 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. $

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


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