Radial solutions for p‐Laplacian Neumann problems with gradient dependence

Minghe Pei, Libo Wang and Xuezhe Lv

Mathematische Nachrichten, 2022, vol. 295, issue 12, 2422-2435

Abstract: We study the radial solutions of the p‐Laplacian Neumann problem with gradient dependence of the type −Δpu=f(|x|,u,|∇u|)inΩ,∂u∂n=0on∂Ω,$$\begin{equation*} {\hspace*{8pc}\left\lbrace \def\eqcellsep{&}\begin{array}{l}-\Delta _{p}u=f(|x|,u,|\nabla u|)\quad \textrm {in} \nobreakspace \Omega ,\\[3pt] \displaystyle \frac{\partial u}{\partial {\bf n}}=0\quad \textrm {on}\nobreakspace \partial \Omega , \end{array} \right.} \end{equation*}$$where Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N(N\ge 2)$ is either a ball or an annulus, and p>1$p>1$. By using the topological transversality method together with the barrier strip technique, we obtain existence results of radial solutions to the above problem, where the nonlinearity f does not need to satisfy any growth restriction.

Date: 2022
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