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arXiv 1206.3196math.AP
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半线性椭圆型方程的一个集中现象

A concentration phenomenon for semilinear elliptic equations

Nils Ackermann, Andrzej Szulkin

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摘要：

对于区域$Ω\subset\dR^N$，我们考虑具有零Dirichlet边界条件的方程$ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$和$p\in(2,2^*)$ .这里，$V\ge 0$和$Q_n$是在$Ω$中包含的区域中为正而在外部为负的有界函数，并且使得集合$\{Q_n>0\}$收缩到点$x_0\inΩ$为$n\to\infty$。我们表明，如果$u_n$是一个非平凡的解决方案，对应于$Q_n$，那么序列$(u_n)$集中在$x_0$相对于$H^1$和某些$L^q$ -规范。我们还表明，如果集合$\{Q_n>0\}$收缩到两个点，$u_n$是基态解，那么它们集中在这些点之一。

英文摘要：

For a domain $Ω\subset\dR^N$ we consider the equation $ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $Ω$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\inΩ$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.

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