Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties. This gives rise to the search for normalised solutions, i.e., when the mass is prescribed. In this talk, I will exploit a novel variational approach, introduced in the context of autonomous Schrödinger equations, to find a least-energy solution to a problem involving the m-Laplacian and a Hardy-type potential. The growth of the non-linearity is mass-supercritical at infinity and at least mass-critical at the origin. An important step in this approach is to show that all the solutions satisfy the Pohozaev identity, which in the presence of a Hardy-type potential was previously known only in the spherical case with m = 1. This talk is based on a joint article with Bartosz Bieganowski and Jaroslaw Mederski, about energy-subcritical non-linearities, and a joint preprint with Bartosz Bieganowski and Olímpio H. Miyagaki, concerning exponential critical non-linear terms in dimension N = 2m.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006