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

Abstract

Abstract In this paper, we study the existence and multiplicity of solutions of the quasilinear problems with minimum and maximum (ϕ(u′(t)))′=(Fu)(t),a.e. t∈(0,T),min{u(t)∣t∈[0,T]}=A,max{u(t)∣t∈[0,T]}=B, $$\begin{aligned}& \bigl(\phi \bigl(u'(t)\bigr)\bigr)'=(Fu) (t),\quad \mbox{a.e. }t\in (0,T), \\& \min \bigl\{ u(t) \mid t\in [0,T]\bigr\} =A, \qquad \max \bigl\{ u(t) \mid t\in [0,T]\bigr\} =B, \end{aligned}$$ where ϕ:(−a,a)→R $\phi :(-a,a)\rightarrow \mathbb{R}$ ( 01 $T>1$ is a constant and A,B∈R $A, B\in \mathbb{R}$ 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.

Keywords

• Mean curvature operators
• Multiplicity
• Minkowski space
• Leray–Schauder degree
• Brosuk theorem