Multiple positive solutions for a quasilinear elliptic equation in $\mathbb {R}^N$

Abstract

In this paper, by means of the extraction of the Palais--Smale sequence in the Nehari manifold, we are concerned with the existence of multiple positive solutions of a class of the p-Laplacian equations involving concave-convex nonlinearities $$\left\{ \begin{array}{ll} -\triangle_p u+|u|^{p-2}u=a(x)|u|^{s-2}u +\lambda b(x)|u|^{r-2}u,\;\;\; x\in {\mathbb R}^N,\\ u\in W^{1,p}({\mathbb R}^N), \end{array} \right.$$ in the whole space ${\mathbb R}^N,$ where $\lambda$ is a positive constant, $1\leq r < p < s < p^*=\frac{Np}{N-p}$, and $a(x)$ and $b(x)$ are nonnegative continuous functions in ${\mathbb R}^N.$