论文标题
涉及非凸范围函数的多重定理
Multiplicity theorems involving functions with non-convex range
论文作者
论文摘要
这是本文证明的结果的一个示例：让$ f：{\ bf r} \ to {\ bf r} $是连续的功能，让$ρ> 0 $，让$ρ> 0 $，然后$ω：[0，ρ[\ to [0，+\ to [0，+\ infty' ρ^ - } \ int_0^ξΩ（x）dx =+\ infty $。考虑$ c^0（[0,1]）\ times c^0（[0,1]）$赋予了规范$$ \ |（α，β）\ | = \ int_0^1 |α（t）| dt+\ int_0^1 |β1|β（t）|β（t） [ - {{\sqrtρ} \ over {2}}，{{\sqrtρ} \ over {2}}}} \ right] $不是常数; $（b）$ for for每个凸集$ s \ subseteq c^0（[[0,1]）\ times c^0（[0,1]）$ c^0（[0,1]）\ times c^0（[0,1]） $ \ case {-Ω\ left（\ int_0^1 | u'（t）|^2dt \ right）两种古典解决方案。
Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf R}$ be a continuous function, let $ρ>0$ and let $ω:[0,ρ[\to [0,+\infty[$ be a continuous increasing function such that $\lim_{ξ\to ρ^-}\int_0^ξω(x)dx=+\infty$. Consider $C^0([0,1])\times C^0([0,1])$ endowed with the norm $$\|(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.
