A fractional Brezis-Nirenberg problem with multipolar Hardy potentials

We study a fractional Brezis–Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points $a_1,\dots ,a_k$, with masses ${\mu }_1,\dots ,{\mu }_k\ge 0$. We assume ${\sum }_{i=1}^k{\mu }_i<{\mathrm{\Lambda }}_{N,s}$ and consider $N>4s$ and $0<{\mu }_{\max }<{\bar{\mu }}_{N,s}$. The interaction between the poles allows existence for $\lambda =0$ and for some negative values of $\lambda$. At a pole $a_j$, this interaction is described by $B_j(\lambda ,\mathbf{a},\mathit{\mu })=\lambda +{\sum }_{i\mathrm{\ne }j}\frac{{\mu }_i}{\mathrm{\mid }{a}_i-a_j{\mathrm{\mid }}^{2s}}$. This leads to a threshold ${\lambda }_{\mathrm{g}\mathrm{e}\mathrm{o}}\le 0$. We prove that the critical Rayleigh quotient is attained for ${\lambda }_{\mathrm{g}\mathrm{e}\mathrm{o}}<\lambda <{\lambda }_1(\mathit{\mu },\mathbf{a})$, and hence the problem admits a positive least-energy solution. If at least two masses are positive, then ${\lambda }_{\mathrm{g}\mathrm{e}\mathrm{o}}<0$, so the result includes $\lambda =0$ and a nonempty interval of negative values of $\lambda$. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.