AbstractIn 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.
Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces
