Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in $\mathbb{R}^{n}$

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of weak solutionsfor nonhomogeneous quasilinear elliptic equations of the form$$-\Delta_n u + a(x)|u|^{n-2}u= b(x)|u|^{n-2}u+g(x)f(u)+\varepsilonh \quad \mbox{in }\mathbb{R}^n ,$$%where $n \geq 2$, $ \Delta_n u \equiv {\rm div}(|\nabla u|^{n-2}\nablau)$ is the $n$-Laplacian and $\varepsilon$ is a positiveparameter. Here the function $g(x)$ may be unbounded in $x$ andthe nonlinearity $f(s)$ has critical growth in the sense ofTrudinger-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 aTrudinger-Moser type inequality, our results are proved by usingEkeland variational principle, minimization and mountain-passtheorem.