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

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$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$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.

Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space

“…Proof. The arguments are quite similar to those from the proof of Proposition 3.2 in [13] (also see [14,Proposition 2]). However, for the sake of completeness, we give a sketch of the proof below.…”

“…This allows us to depict multiplicity and localization informations about solutions. We note that our lower and upper solutions method -which is different from the one in [16,17] used for radial systems, as well as the multiplicity and localization results we obtain for problem (1) (Propositions 3.3 -3.5) are inspired by the corresponding ones proved for a single equation in [13,14].…”

“…, which together with (20) and the additivity-excision property of the Leray-Schauder degree, imply (see [13,14])…”

“…In recent years, many papers were devoted to the study of Dirichlet problems for a single equation with operator M in a general bounded domain (see e.g. [3,5,6,13,14,15,20]), while at our best knowledge, for systems with such an operator the study was recently initiated in [18]. Also, results for systems having a radial structure in a ball were obtained in [4,16,17].…”

“…The quasi-monotonicity of f 1 and f 2 will play a key role here. Adopting some ideas from [13,14], we divide the proof in three parts.…”