非径向区域上超临界椭圆问题的非光滑变分方法
Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach
摘要：
本文研究了\begin{equation} \label{eqx} \left\{\begin{array}{ll} -Δu = a(x) u^{p-1} & \mbox{ in } Ω, \\ u>0 & \mbox{ in } Ω, \\ u= 0 & \mbox{ on } \pOm, \end {array}\right. \end{equation}的正古典解，其中$Ω$是$\IR^N$ $(N \ge3)$中的有界环形区域（不一定是环），而$ a(x)$是非负连续函数.当问题具有一定的对称性和单调性条件时，我们证明了$p$在一定范围的超临界值下经典正解的存在性。作为结果的一个结果，我们将证明当$ a(x)=1$和$Ω$是一个具有一定半径假设的环时，（\ref{eqx}）有$\Bigl\lfloor\frac{N}{2} \Bigr\rfloor$（$\frac{N}{2}$的底）正的非径向解.在环形区域的情况下，我们也得到了正解的存在性。我们的方法是基于一个新的变分原理，允许一个处理超临界问题变分限制相应的功能上的一个适当的凸子集，而不是整个空间在一个温和的不变性财产为代价。
英文摘要：
In this paper we are interested in positive classical solutions of \begin{equation} \label{eqx} \left\{\begin{array}{ll} -Δu = a(x) u^{p-1} & \mbox{ in } Ω, \\ u>0 & \mbox{ in } Ω, \\ u= 0 & \mbox{ on } \pOm, \end {array}\right. \end{equation} where $Ω$ is a bounded annular domain (not necessarily an annulus) in $\IR^N$ $(N \ge3)$ and $ a(x)$ is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of $p$ when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall show that (\ref{eqx}) has $\Bigl\lfloor\frac{N}{2} \Bigr\rfloor$ (the floor of $\frac{N}{2}$) positive nonradial solutions when $ a(x)=1$ and $Ω$ is an annulus with certain assumptions on the radii. We also obtain the existence of positive solutions in the case of toroidal domains. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space at the expense of a mild invariance property.
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