Quasilinear elliptic equations in R^N via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem \begin{equation*}\label{eq} \left\{ \begin{array}{ll} -\n \cdot \left[\phi'(|\n u|^2)\n u \right] +|u|^{\a-2}u =|u|^{s-2} u, & x\in \RN, \\ u(x) \to 0 , \quad \hbox{as }|x|\to \infty, \end{array} \right. \end{equation*} where $N\ge 2$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $1< p<q<N$, $1<\a\le p^* q'/p'$ and $\max\{q,\a\}< s<p^*$, being $p^*=\frac{pN}{N-p}$ and $p'$ and $q'$ the conjugate exponents, respectively, of $p$ and $q$. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.