Concave-convex effects for critical quasilinear elliptic problems

TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of positive solutions for a quasilinear elliptic problem involving the -Laplacian operator with a concave-convex nonlinearity exhibiting critical growth, using variational methods.

Contribution

It establishes new results on positive solutions for a critical quasilinear elliptic problem with concave-convex nonlinearities, employing the Nehari method and concentration compactness.

Findings

Existence of positive solutions including ground state.

Multiple solutions under certain conditions.

Asymptotic behavior characterized using variational techniques.

Abstract

It is established existence, multiplicity and asymptotic behavior of positive 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 semilinear term in the quasilinear equation 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.