On a fractional degenerate Kirchhoff-type problem

Abstract

In this paper, we study a highly nonlocal parametric problem involving a fractional-type operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Our approach is of variational nature; by working in suitable fractional Sobolev spaces which encode Dirichlet homogeneous boundary conditions, and exploiting an abstract critical point theorem for smooth functionals, we derive the existence of at least three weak solutions to our problem for suitable values of the parameters. Finally, we provide a concrete estimate of the range of these parameters in the autonomous case, by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. The methods adopted here can be exploited to study different classes of elliptic problems in presence of a degenerate nonlocal term.