<p>In this paper, we are concerned with the Dirichlet problem of quasilinear differential system, involving the mean curvature operator in Minkowski space <Equation ID="Equ29"> <EquationSource Format="TEX">\(\begin{aligned} \mathcal {M}(w)=\left( \frac{w'}{\sqrt{1-|w'|^{2}}}\right) '. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">M</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced close=")" open="("> <mfrac> <msup> <mi>w</mi> <mo>′</mo> </msup> <msqrt> <mrow> <mrow> <mn>1</mn> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi>w</mi> <mo>′</mo> </msup> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mfenced> <mo>′</mo> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Using global bifurcation technique, we obtain the existence of an unbounded branch of positive solutions, which is unbounded in the positive <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-direction. We also establish a Calabi-Bernstein type asymptotic property for a pair of one-signed solutions on certain bifurcation branch converge to a piecewise linear function. The proof of our main results is based upon the global bifurcation theory and the Sturm separation theorem.</p>