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 \(\rho>0\) and let \(\omega:[0,\rho[\to [0,+\infty[\) be a continuous increasing function such that \(\lim_{t\to \rho^-}\omega(t)=+\infty\). Consider \(C^0([0,1])\times C^0([0,1])\) endowed with the norm

$$\|(\alpha,\beta)\|=\int_0^1|\alpha(t)|dt+\int_0^1|\beta(t)|dt\ .$$

Then, the following assertions are equivalent:

(a) the restriction of \(f\) to \(\left [-{{\sqrt{\rho}}\over {2}},{{\sqrt{\rho}}\over {2}}\right ]\) 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 \((\alpha,\beta)\in S\) such that the problem

$$\cases{-\omega\left(\int_0^1|u'(t)|^2dt\right)u''=\beta(t)f(u)+\alpha(t) & in $[0,1]$\cr & \cr u(0)=u(1)=0\cr & \cr

\int_0^1|u'(t)|^2dt<\rho\cr}$$

has at least two classical solutions.