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# Bifurcation results for a fractional elliptic equation with critical exponent in R^n

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### Abstract

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.

### Author and article information

###### Journal
2014-10-12
2015-08-04
1410.3076