Quasilinear elliptic equations in mathbb{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 $$begin{aligned} left{ begin{array}{l@{quad }l} -nabla cdot left[phi ^{prime }(|nabla u|^2)nabla u right] +|u|^{alpha -2}u =|u|^{s-2} u,&xin mathbb{R }^{N}, u(x) rightarrow 0, quad text{ as} |x|rightarrow infty , end{array} right. end{aligned}$$ where $Nge 2, phi (t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t, 1< p and $max {q,alpha }< s being $p^*=frac{pN}{N-p}$ and $p^{prime }$ and $q^{prime }$ 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.