Global bifurcation for problem with mean curvature operator on general domain

Abstract

We establish the existence of nontrivial nonnegative solution for the following 0-Dirichlet problem with mean curvature operator in the Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\text {in}\,\, \Omega ,\\ u=0&{}\text {on}\,\, \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a general bounded domain of $$\mathbb {R}^N$$ . By bifurcation and topological methods, we determine the interval of parameter $$\lambda $$ in which the above problem has nontrivial nonnegative solution according to sublinear or linear nonlinearity at zero.