Abstract:

In this paper, we study the following quasilinear nonhomogeneous elliptic equations of the form

$-\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N}, $

where $1 <p\leq N$ ($N\geq 3$), $\frac{1}{2} <\alpha\leq 1 $, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $h \in C(\mathbb{R}, \mathbb{R})$ and $g\in L^{p'}(\mathbb{R}^{N})$, where $p'=\frac{p}{p-1}$, is a disturbance term. Using a variable replacement and minimax method, we show the existence and multiplicity of solutions to this problem.

Key words: Quasilinear elliptic equations, Ekeland's variational principle, Palais-Smale sequences