Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Abstract

The multiple results of positive solutions for the following quasilinearelliptic equation: $-{\Delta }_{p}u=\lambda f\left(x\right){\left|u\right|}^{q-2}u+g\left(x\right){\left|u\right|}^{p^{\ast }-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, are established. Here, $0\in \Omega$ is a bounded smooth domain in ${ℝ}^N,{\Delta }_p$ denotes the $p$-Laplacian operator, $1\le q<p<N,p^{\ast }=Np/\left(N-p\right),\lambda$ is a positive real parameter, and $f,g$ are continuous functions on $\overline{\Omega }$ which are somewhere positive but which may change sign on $\Omega$. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.