article

Title: On the number of positive solutions of a quasilinear elliptic problem

Authors: Marcelo Furtado and Giovany Malcher Figueiredo

Issue: Volume 55 (2006), Issue 6, 1835-1856

Abstract:

$We obtain multiplicity of positive solutions for the quasilinear equation \[ {-}\varepsilon^{p} \div(a(x)|\nabla u|^{p-2} \nabla u) + u^{p-1} = f(u) \quad\mbox{in }\mathbb{R}^{N}, \quad u \in W^{1,p}(\mathbb{R}^{N}), \] where $\varepsilon > 0$ is a small parameter, $1 < p < N$, $f$ is a subcritical nonlinearity and $a$ is a positive potential such that $\inf_{\partial\Lambda}a > \inf_{\Lambda}a$ for some open bounded subset $\Lambda \subset \mathbb{R}^{N}$. We relate the number of positive solutions with the topology of the set where $a$ attains its minimum in $\Lambda$. The result is proved by using Ljusternik-Schnirelmann theory.$