Combined effects in Kirchhoff fractional elliptic problems with lack of compactness

The abstract is focused on recent results concerning existence, multiplicity and asymptotic behavior of positive solutions of some Kirchhoff type problems, involving fractional integro-differential elliptic operators and presenting also difficulties due to intrinsic lacks of compactness, which arise from different reasons. The problems presented are highly nonlocal because of the presence of the fractional integro-differential elliptic operators and of the Kirchhoff coefficients. The proof techniques should therefore overcome the nonlocal nature of the problems as well as the lack of compactness, and the suitable strategies adopted depend of course on the problem under consideration.