论文标题
具有多个全局最小值的一类功能
A class of functionals possessing multiple global minima
论文作者
论文摘要
我们为梯度系统获得了新的多重结果。这是一个非常特殊的推论：令$ω\ subset {\ bf r}^n $（$ n \ geq 2 $）是一个平稳的界面域，让$φ：{\ bf r}^2 \ to {\ bf r} $ a $ c^1 $ c^1 $ a $ c^1 $函数，$ c _（$ m（up）， {\ bf r}^2} {{|φ_U（u，v）|+|+|φ_v（u，v）|} \ over {1+ | u |^p+|^p+|^p}} <+\ iffty $ p}然后，对于每个凸集，$ s \ subseteq l^{\ infty}（ω）\ times l^{\ infty}（ω）$ lende in $ l^2（ω）\ times l^2（ω）$，存在$（α，β）\ in S $中的问题 $ \ case {-ΔU=（α（x）\ cos（φ（u，v）） - β（x）\ sin（φ（u，v）））φ_u（u，v）＆in $ \ cr＆\ cr＆\ cr-uccr-ΔV= （α（x）\ cos（φ（u，v）） - β（x）\ sin（φ（u，v）））φ_v（u，v）＆in $ω$ \ cr＆\ cr＆\ cr＆\ cr＆\ cr＆\ cr u = v = 0＆on $ \partialΩ$ \ cr} $ \ cr} $ 4至少有三个薄弱的解决方案，其中两个是$ h^$ h^$ h^1_0 in $ h^h^1_0（功能$$（u，v）\ to {{1} \ aver {2}}} \ left的h^1_0（ω）$ $$ - \int_Ω（α（x）\ sin（φ（u（x），v（x））））+β（x）\ cos（φ（u（x），v（x））））dx \ $ $
We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $Ω\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $Φ:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $Φ(0,0)=0$, such that $$\sup_{(u,v)\in {\bf R}^2}{{|Φ_u(u,v)|+|Φ_v(u,v)|}\over {1+|u|^p+|v|^p}}<+\infty$$ where $p>0$, with $p={{2}\over {n-2}}$ when $n>2$. Then, for every convex set $S\subseteq L^{\infty}(Ω)\times L^{\infty}(Ω)$ dense in $L^2(Ω)\times L^2(Ω)$, there exists $(α,β)\in S$ such that the problem $$\cases {-Δu=(α(x)\cos(Φ(u,v))-β(x)\sin(Φ(u,v)))Φ_u(u,v) & in $Ω$ \cr & \cr -Δv= (α(x)\cos(Φ(u,v))-β(x)\sin(Φ(u,v)))Φ_v(u,v) & in $Ω$ \cr & \cr u=v=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)\times H^1_0(Ω)$ of the functional $$(u,v)\to {{1}\over {2}}\left ( \int_Ω|\nabla u(x)|^2dx+\int_Ω|\nabla v(x)|^2dx\right )$$ $$-\int_Ω(α(x)\sin(Φ(u(x),v(x)))+β(x)\cos(Φ(u(x),v(x))))dx\ .$$
