Bifurcation results for a fractional elliptic equation with critical exponent in R^n
Bifurcation results for a fractional elliptic equation with critical exponent in R^n
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发表时间：
2014-10
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通讯作者：
S. Dipierro;María Medina;I. Peral;E. Valdinoci
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作者：
S. Dipierro;María Medina;I. Peral;E. Valdinoci
本文研究了在$\R^n$中作为分数阶临界Sobolev指数问题的摄动得到的一些非线性椭圆方程，即$$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {{in}}\R^n,$$中$s\in(0,1)$, $n>4s$, $\epsilon>0$是一个小参数，$p=\frac{n+2s}{n-2s}$, $0<q<p$, $h$是一个连续紧支持函数。为了构造这个方程的解，我们使用Lyapunov-Schmidt约简，它利用了问题的变分结构。对于这种情况，$0<q<1$特别困难，由于缺乏相关能量泛函的正则性，我们需要引入新的泛函设置并发展适当的分数阶椭圆正则性理论。
In this paper we study some nonlinear elliptic equations in $\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {{in}}\R^n,$$ where $s\in(0,1)$, $n>4s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.