论文标题
一种新型的淋巴结解决方案，用于具有超临界增长的椭圆方程
A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth
论文作者
论文摘要
在本文中，我们旨在调查以下单一扰动的椭圆问题$$ \ weft \ { \ begin {array} {ll} \ displayStyle- \ varepsilon^2 \ triangle {u}+| x |^ηu= | x |^|^|^ηf（u）＆\ mbox {in} \，\，\，a，a，a， u = 0＆\ mbox {on} \，\，\ partial a， \ end {array} \ right。 $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth.通过减少参数，我们表明存在一个nodal解决方案$ u_ \ e $，恰好有两个正峰和两个负峰，它们集中在两个尺寸$ n-1 $的两个不同的正交球上，为$ \ e \ e \ e rightarrow0 $。特别是，当参数$η> 2 $，$η= 2 $和$η<2 $时，我们建立了四个峰的不同浓度现象。
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^ηu =|x|^ηf(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension $N-1$ as $\e\rightarrow0$. In particular, we establish different concentration phenomena of four peaks when the parameter $η>2$, $η=2$ and $η<2$.
