We mainly study tangentiality, Lyapunov-type inequalities, bifurcation, existence, multiplicity of positive solutions and nodal solutions for problems concerning with the mean curvature operator on bounded or exterior domains. The results are contained in the following four chapters.In Chapter 3, we analyze non $\frac{\pi}{4}$-tangentiality or $\frac{\pi}{4}$-tangentiality of solutions for several types of scalar equations and systems of one-dimensional mean curvature problems in Minkowski space with singular weight functions. Moreover, as applications of non $\frac{\pi}{4}$-tangential solutions, we estimate Lyapunov-type inequalities for scalar equation as well as systems. We also estimate Lyapunov-type inequality for a scalar equation with singular coefficient in an operator.In Chapter 4, we study the existence of positive radial solutions for a mean curvature problem in Minkowski space on an exterior domain. Based on C¹-regularity of solutions, which is closely related to the property of nonlinearity $f$ near 0, and by using the global bifurcation theory, we establish some existence results when $f$ is sublinear at ∞.In Chapter 5, we investigate the existence and multiplicity of nodal radial solutions for a mean curvature problem in Minkowski space on an exterior domain of a ball using the global bifurcation theory. Moreover, we establish the asymptotic behavior of solutions on the subcontinua which are bifurcating from a trivial branch. All results are obtained by considering a radially equivalent problem with a singular weight.In Chapter 6, applying bifurcation method, we investigate the global structures of solutions for a one-dimensional mean curvature problem in Minkowski space under different behaviors of nonlinear $f$ near zero, {\it i.e.}, linear, superlinear, and sublinear, respectively.