Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball

Abstract

By using a shooting technique, we prove that the quasilinear boundary value problem $$\text{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u{|}^2}}\right)+\lambda q(|x|)|u{|}^{p-1}u=0,u{|}_{\partial \mathcal{B}}=0,$$ where $\mathcal{B}\subset {\mathbb{R}}^N$ is a ball and $p>1$, has more and more pairs of nodal solutions on growing of the parameter $\lambda >0$. The radial Neumann problem and the periodic problem for the corresponding one-dimensional equation are considered, as well.