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Abstract. Aim of this paper is to give the details of the proof of some density properties of smooth and compactly supported functions in the fractional Sobolev spaces and suitable modifications of them, which have recently found application in variational problems. The arguments are rather technical, but, roughly speaking, they rely on a basic technique of convolution (which makes functions C ∞ ), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.

Abstract. This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problemwhere Ω ⊂ R n is a bounded domain, 2 * s is the critical exponent of the fractional Sobolev space H s (R n ), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

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