Multiple positive solutions to a class of quasi-linear elliptic equations involving critical Sobolev exponent (Q398690)

A quasi-linear elliptic problem involving two parameters\(f,g\) is considered, related with the \(p\)-Laplacian. An energy functional \(J\) is introduced, whose critical points are weak solutions of the considered problem. The authors study the properties of \(J\). The main tools are Nehari manifolds, the mini-max principle and the Palais -Smale sequences and condition. The analysis of the properties of \(J\) is giving the existence of two or three positive solutions of the considered problem, if the parameters \(f,g\) are verifying some conditions. This is an improvement of the results of \textit{T.-S. Hsu} [Abstr. Appl. Anal. 2009, Article ID 652109, 24 p. (2009; Zbl 1179.35134)], who used also the fibering maps method.