Existence and Nonexistence of Solutions of Minkowski-Curvature Problems in Exterior Domains

Abstract

In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: $$ \begin{cases} \ -\text{div} \big(\phi(\nabla v(x))\big)=k(x)f(v(x)), \quad\quad x\in\Omega,\\ \ v=0\ \text{on} \ \partial\Omega, \qquad\lim\limits_{x\rightarrow\infty}v(x)=0\\ \end{cases} $$ for R sufficiently large, where $\phi(s)=\frac{s}{\sqrt{1-s^{2}}}$ for $s\in{\mathbb R}$ with $s^2\lt1,$ $\Omega=\{x\in{{\mathbb R}^{N}}:\ |x| \gt R\}$, $N\geq3$ is an integer, $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^{N}$, R is a positive parameter, $f:\mathbb{R}\rightarrow\mathbb{R}$ is an odd and locally Lipschitz continuous function and $k \in C^{1}(\mathbb{R}^{+},\ \mathbb{R}^{+})$ with $\mathbb{R}^{+}=(0, +\infty)$. We also apply the fixed-point index theory to establish the existence of positive radial solutions of the above problems for R sufficiently small.