Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in R^n

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of weak solutions for nonhomogeneous quasilinear elliptic equations of the form -\Delta_n u + a(x)|u|^{n-2}u= b(x)|u|^{n-2}u+g(x)f(u)+\varepsilon h \quad \mbox{in }\mathbb{R}^n , where $n \geq 2$, $ \Delta_n u \equiv \dive(|\nabla u|^{n-2}\nabla u)$ is the $n$-Laplacian and $\varepsilon$ is a positive parameter. Here the function $g(x)$ may be unbounded in $x$ and the nonlinearity $f(s)$ has critical growth in the sense of Trudinger-Moser inequality, more precisely $f(s)$ behaves like $e^{\alpha_0 |s|^{n/(n-1)}}$ when $s\to+\infty$ for some $\alpha_0>0$. Under some suitable assumptions and based on a Trudinger-Moser type inequality, our results are proved by using Ekeland variational principle, minimization and mountain-pass theorem.