Abstract We show the existence and multiplicity of radial solutions for the problems with minimum and maximum involving mean curvature operators in the Minkowski space: { div ⁡ ( ϕ N ⁢ ( ∇ ⁡ v ) ) = F ⁢ ( v ) ⁢ ( | x | )   for a.e. ⁢ R 1 < | x | < R 2 , x ∈ ℝ N , N ≥ 2 , min ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = A , max ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = B , $\left\{\begin{aligned} &\displaystyle\operatorname{div}(\phi_{N}(\nabla v))=F(% v)(\lvert x\rvert)\quad\text{for a.e. }R_{1}<\lvert x\rvert<R_{2},\,x\in% \mathbb{R}^{N},\,N\geq 2,\\ &\displaystyle\min\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}% =A,\quad\max\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}=B,% \end{aligned}\right.$ where ϕ N ⁢ ( z ) = z / 1 - | z | 2 ${\phi_{N}(z)=z/\sqrt{1-\lvert z\rvert^{2}}}$ , z ∈ ℝ N ${z\in\mathbb{R}^{N}}$ , R 1 , R 2 , A , B ∈ ℝ ${R_{1},R_{2},A,B\in\mathbb{R}}$ are constants satisfying 1 < R 1 < R 2 - 1 ${1<R_{1}<R_{2}-1}$ and A < B ${A<B}$ ; | ⋅ | ${\lvert\cdot\rvert}$ denotes the Euclidean norm in ℝ N ${\mathbb{R}^{N}}$ , and F : C 1 ⁢ [ R 1 , R 2 ] → L 1 ⁢ [ R 1 , R 2 ] ${F:C^{1}[R_{1},R_{2}]\to L^{1}[R_{1},R_{2}]}$ is an unbounded operator. By using the Leray–Schauder degree theory and the Borsuk theorem, we prove that the problem has at least two different radial solutions.