Ambrosetti–Prodi type results for a Neumann problem with a mean curvature operator in Minkowski spaces

Abstract

This paper is concerned with the existence and multiplicity of solutions for the following Neumann problems with mean curvature operator in the Minkowski space:

$${\left(\frac{u^{\prime }}{\sqrt{1-{u{}^{\prime }}^2}}\right)}^{\prime }+a\left(x\right)g\left(u\right)=\mu +p\left(x\right),x\in \left(0,T\right),u^{\prime }\left(0\right)=0=u^{\prime }\left(T\right),$$

where $a,p\in L^{\infty }\left(0,T\right)$, $\mu \in ℝ$, and $g\in C^1\left(ℝ\right)$ satisfies the coercivity condition $g\left(u\right)\to +\infty$ as $\left|u\right|\to +\infty$. We show the existence results of two solutions in terms of the value of the parameter $\mu$ via the shooting method.