Dirichlet problems with mean curvature operator in Minkowski space (Q2829828)

This paper is a survey of results obtained by the authors concerning the existence and multiplicity of solutions for the following Dirichlet problems involving the mean curvature operator in Minkowski space: NEWLINE\[NEWLINE \operatorname{div}\biggl(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\biggr)+\lambda \varphi(|x|)v^q=0, \text{ in }\mathcal{B}(R):=\{x\in \mathbb{R}^N: |x|<R\},\, u=0\text{ on }\partial\mathcal{B}(R)\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\operatorname{div}\biggl(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\biggr)+f(x,u)=0,\text{ in }\Omega,\quad u=0\text{ on }\partial \Omega,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\operatorname{div}\biggl(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\biggr)\in [\underline{f}(x,u),\overline{f}(x,u)],\text{ in }\Omega,\quad u=0\text{ on }\partial \Omega,\tag{3}NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\lambda>0\), \(q>1\), \(\varphi:[0,+\infty)\rightarrow [0,+\infty)\) is a continuous function with \(\varphi(0)=0\) and \(\varphi(t)>0\) for \(t>0\), \(f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}\) is a measurable function, and \(\underline{f}(x,t)=\lim_{\delta \searrow 0}\mathrm{essinf}\{f(x,s): \;|t-s|<\delta\}\), \(\overline{f}(x,t)=\lim_{\delta \searrow 0}\mathrm{esssup}\{f(x,s): \;|t-s|<\delta\}\).NEWLINENEWLINEFor the entire collection see [Zbl 1345.35003].