Raffaella Servadei
The Yamabe equation in a non-local setting
(85K, LaTeX)
ABSTRACT. Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator~$\mathcal L_K$
$$\left\{
egin{array}{ll}
\mathcal L_K u+\lambda u+|u|^{2^*-2}u=0 & \mbox{in } \Omega\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}
ight.$$
where $s\in (0,1)$, $\Omega$ is an open bounded set of $\RR^n$, $n>2s$, with Lipschitz boundary, $\lambda$ is a positive real parameter, $2^*=2n/(n-2s)$ is a fractional critical Sobolev exponent, while $\mathcal L_K$ is the non-local integrodifferential operator
$$\mathcal L_Ku(x)=
rac12
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,.$$
As a concrete example, we consider the case when $K(x)=|x|^{-(n+2s)}$\,, which gives rise to the fractional Laplace operator $-(-\Delta)^s$\,.