Critical quasilinear elliptic problems using concave–convex nonlinearities

Abstract

It is established existence, multiplicity, and asymptotic behavior of nonnegative solutions for a quasilinear elliptic problem driven by the $$\Phi $$ -Laplacian operator. One of these solutions is obtained as ground-state solution by applying the well-known Nehari method. The nonlinear term is a concave–convex function which presents a critical behavior at infinity. The concentration-compactness principle is used in order to recover the compactness required in variational methods.