### Multiplicity theorems involving functions with non-convex range

Biagio Ricceri

#### 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.

#### Keywords

minimax; global minimum; multiplicity; non-convex sets; Chebyshev sets; Kirchhoff-type problems

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DOI: http://dx.doi.org/10.24193/subbmath.2023.1.09

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