12-40 Raffaella Servadei

The Yamabe equation in a non-local setting (85K, LaTeX) Apr 25, 12

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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 $-(-\Delta)^s$\,.

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