论文标题
具有三个解决方案的一类方程
A class of equations with three solutions
论文作者
论文摘要
这是本文获得的结果之一：令$ω\ subset {\ bf r}^n $是一个平稳的界面域，让$ q> 1 $，带$ q <{{n+2} \ fover {n+2} \ fover {n-2}} $ n \ geq 3 $，如果$ n \ geq 3 $，而让$λ_1$ case $λ_1$ $ $ eigenvalue和Δ $ω$ \ cr＆\ cr u = 0＆on $ \partialΩ$ \。\ cr} $$，然后，对于每一个$λ>λ_1$，对于每一个coNvex set $ s \ subseteq l^{\ infty}（\ infty}（ω）$ dence in $ l^2（ω）$ $ \ case {-ΔU=λ（u^+ - （u^+）^q）+α（x）＆in $ω$ \ cr＆\ cr＆\ cr＆\ cr＆\ cr u = 0＆on $ \partialΩ$ \ cr} $ \ cr} $ \ cr} $ \ cr} $ \ cr}至少具有三个弱解决方案，其中两个是$ h^1_0 $}的全局最小值（均超过$ h^1_0（ω）$}。 {2}} \int_Ω| \ nabla u（x）|^2dx-λ\int_Ω\ left（{{{1} \ over {2}}} | ）dx- \int_Ωα（x）u（x）dx \ $$其中$ u^+= \ max \ {u，0 \} $。
Here is one of the results obtained in this paper: Let $Ω\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $λ_1$ be the first eigenvalue of the problem $$\cases{-Δu=λu & in $Ω$ \cr & \cr u=0 & on $\partialΩ$\ .\cr}$$ Then, for every $λ>λ_1$ and for every convex set $S\subseteq L^{\infty}(Ω)$ dense in $L^2(Ω)$, there exists $α\in S$ such that the problem $$\cases{-Δu=λ(u^+-(u^+)^q)+α(x) & in $Ω$ \cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)$ of the functional $$u\to {{1}\over {2}}\int_Ω|\nabla u(x)|^2dx-λ\int_Ω\left ({{1}\over {2}}|u^+(x)|^2-{{1}\over {q+1}}|u^+(x)|^{q+1}\right )dx-\int_Ωα(x)u(x)dx\ $$ where $u^+=\max\{u,0\}$.
