The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Abstract

Abstract We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space { - div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) = f ⁢ ( x , u , ∇ ⁡ u ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω . $\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$ The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.