Multiplicity of positive solutions for a class of quasilinear problems (Q1038572)

The paper is concerned with the existence of multiple positive solutions for a quasilinear elliptic problem. The study can be seen as a complement of the study made in some classical and recent papers because in the papers where the operator \[ L_p u= \varepsilon^p \Delta_p u+ \varepsilon^p \Delta(u^2)u,\quad 2\leq p> N, \] has been studied (\(\Delta_p\) is the \(p\)-Laplacian operator and \(\varepsilon\) is a small parameter), the existence of multiple positive solutions was not considered by employing the Lusternick-Schnirelman category. We point out that a crucial point to applying Lusternick-Schnirelman category is to prove the well-known Palais-Smale condition.