On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator

Abstract

In this paper, we are concerned with necessary conditions for the existence of positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space \begin{equation*} \begin{aligned} -\text{div}\left(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\right)&=f(u) &\quad &\text{in}\ \Omega,\\ u&=0 &\quad &\text{on}\ \partial \Omega, \end{aligned} \end{equation*} whose supremum norm bears a certain relationship to zeros of the nonlinearity $f$.

Keywords

• dirichlet problem
• minkowski-curvature operator
• positive solutions
• lower and upper solutions