Radial Solutions for p-Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions

Abstract

We study the existence of radial solutions for the p-Laplacian Neumann problem with gradient term of the type $$\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p}u=f(|x|,u,x\cdot \nabla u)\quad \text {in} ~\varOmega ,\\ \displaystyle \frac{\partial u}{\partial \mathbf{n} }=0\quad \text {on}~ \partial \varOmega , \end{array} \right. \end{aligned}$$ where $$\Delta _pu=\text {div}(|\nabla u|^{p-2}\nabla u)$$ is the p-Laplace operator with $$p>1$$ , $$\varOmega \subset \mathbb {R}^N(N\ge 2)$$ is a ball. We do not impose any growth restrictions on the nonlinearity. By using the topological transversality method together with the barrier strip technique, the existence of radial solutions to the above problem is obtained.