  Eringen's model is one of the most popular theories in nonlocal elasticity.
It has been applied to many practical situations with the objective of removing
the anomalous stress concentrations around geometric shape singularities, which
appear when the local modelling is used. Despite the great popularity of
Eringen's model in mechanical engineering community, even the most basic
questions such as the existence and uniqueness of solutions have been rarely
considered in the research literature for this model. In this work we focus on
precisely these questions, proving that the model is in general ill-posed in
the case of smooth kernels, the case which appears rather often in numerical
studies. We also consider the case of singular, non-smooth kernels, and for the
paradigmatic case of the Riesz potential we establish the well-posedness of the
model in fractional Sobolev spaces. For such a kernel, in dimension one the
model reduces to the well-known fractional Laplacian. Finally, we discuss
possible extensions of Eringen's model to spatially heterogeneous material
distributions.
