Multiplicity of solutions for mean curvature operators with minimum and maximum in Minkowski space

In this paper, we study the existence and multiplicity of solutions of the quasilinear problems with minimum and maximum $$\begin{aligned}& \bigl(\phi \bigl(u'(t)\bigr)\bigr)'=(Fu) (t),\quad \mbox{a.e. }t\in (0,T), \\& \min \bigl\{ u(t) \mid t\in [0,T]\bigr\} =A, \qquad \max \bigl\{ u(t) \mid t\in [0,T]\bigr\} =B, \end{aligned}$$ (ϕ(u′(t)))′=(Fu)(t),a.e. t∈(0,T),min{u(t)∣t∈[0,T]}=A,max{u(t)∣t∈[0,T]}=B, where $\phi :(-a,a)\rightarrow \mathbb{R}$ϕ:(−a,a)→R ($0< a<\infty $0<a<∞) is an odd increasing homeomorphism, $F:C^{1}[0,T]\rightarrow L^{1}[0,T]$F:C1[0,T]→L1[0,T] is an unbounded operator, $T>1$T>1 is a constant and $A, B\in \mathbb{R}$A,B∈R satisfy $B>A$B>A. By using the Leray–Schauder degree theory and the Brosuk theorem, we prove that the above problem has at least two different solutions.