Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight

TL;DR

This paper establishes the existence of two positive radial solutions for a Minkowski-curvature Neumann problem with an indefinite weight function, using degree theory and extending previous results to broader nonlinearities.

Contribution

It proves the existence of positive solutions for a nonlinear PDE with indefinite weight, employing Leray-Schauder degree theory and extending to more general nonlinearities.

Findings

Existence of a pair of positive radial solutions for the problem.

Results hold for large parameter ; .

Extension to a broader class of nonlinearities.

Abstract

We prove the existence of a pair of positive radial solutions for the Neumann boundary value problem \begin{equation*} \begin{cases} \, \mathrm{div}\,\Biggl{(} \dfrac{\nabla u}{\sqrt{1- | \nabla u |^{2}}}\Biggr{)} + \lambda a(|x|)u^p = 0, & \text{in $B$,} \\ \, \partial_{\nu}u=0, & \text{on $\partial B$,} \end{cases} \end{equation*} where $B$ is a ball centered at the origin, $a(|x|)$ is a radial sign-changing function with $\int_B a(|x|)\,\mathrm{d}x < 0$, $p>1$ and $\lambda > 0$ is a large parameter. The proof is based on the Leray-Schauder degree theory and extends to a larger class of nonlinearities.