Multiplicity theorems involving functions with non-convex range

Abstract

Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf R}$ be a continuous function, let $ρ>0$ and let $ω:[0,ρ[\to [0,+\infty[$ be a continuous increasing function such that $\lim_{ξ\to ρ^-}\int_0^ξω(x)dx=+\infty$. Consider $C^0([0,1])\times C^0([0,1])$ endowed with the norm $$\|(α,β)\|=\int_0^1|α(t)|dt+\int_0^1|β(t)|dt\ .$$ Then, the following assertions are equivalent: $(a)$ the restriction of $f$ to $\left [-{{\sqrtρ}\over {2}},{{\sqrtρ}\over {2}}\right ]$ is not constant; $(b)$ for every convex set $S\subseteq C^0([0,1])\times C^0([0,1])$ dense in $C^0([0,1])\times C^0([0,1])$, there exists $(α,β)\in S$ such that the problem $$\cases{-ω\left(\int_0^1|u'(t)|^2dt\right)u"=β(t)f(u)+α(t) & in $[0,1]$\cr & \cr u(0)=u(1)=0\cr & \cr \int_0^1|u'(t)|^2dt<ρ\cr}$$ has at least two classical solutions.