Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain

Abstract

We are concerned with the problem with Minkowski-curvature operator on an exterior domain $ \begin{align} \left\{\begin{array}{ll} -\text{div}\Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) = \lambda K(|x|)\frac{f(u)}{u^\gamma}\ \ \ &\text{in}\ B^c,\\[2ex] \frac{\partial u}{\partial n}|_{\partial B^c} = 0, \ \ \lim\limits_{|x|\to\infty}u(x) = 0, \end{array} \right. \end{align} \quad\quad\quad\quad\quad (P) $ where $ 0\leq\gamma < 1 $, $ B^c = \{x\in \mathbb{R}^N: |x| > R\} $ is a exterior domain in $ \mathbb{R}^N $, $ N > 2 $, $ R > 0 $, $ K\in C([R, \infty), (0, \infty)) $ is such that $ \int_R^\infty rK(r)dr < \infty $, the function $ f:[0, \infty)\to (0, \infty) $ is a continuous function such that $ \lim\limits_{s\to\infty}\frac{f(s)}{s^{\gamma+1}} = 0 $ and $ \lambda > 0 $ is a parameter. We show that problem $ (P) $ has at least one positive radial solution for all $ \lambda > 0 $. The proof of our main result is based upon the method of sub and super solutions.

Keywords

• mean curvature operator
• exterior domain
• singular
• positive radial solutions
• sub and super solutions