Existence results for a critical fractional equation

We are concerned with existence results for a critical problem of Brezis-Nirenberg-type driven by an integro-differential operator of fractional nature. The latter includes, for a specific choice of the kernel, the usual fractional Laplacian. Under mild assumptions on the subcritical part of the nonlinearity, we provide first the existence of one weak solution through direct minimization of the energy in a small ball of a certain fractional Sobolev space. This approach remains still valid when adding small singular terms. We finally show that for appropriate choices of the parameters involved the mountain-pass approach is also applicable and yields another existence result.