The Brézis–Nirenberg equation for the $ {(m,p)} $ Laplacian in the whole space

In this paper, we prove the existence of a nontrivial (weak) solution of the Brézis–Nirenberg equation for the $ (m, p) $ Laplacian in the whole space $ \mathbb R^N $. Critical problems have been intensively studied in the last decades, starting with the pioneering paper by Brézis and Nirenberg for the Dirichlet Laplacian problems in bounded domains of $ \mathbb R^N $. Even if we obtain existence of solutions for the $ (m, p) $ Laplacian Brézis–Nirenberg equation in the whole space via standard variational techniques, the last step of the proof is pretty involved and requires new ideas and a delicate use of the Talenti functions.