Advances in Difference Equations (Dec 2019)
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.
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