In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.

In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.