Existence of radial bounded solutions for some quasilinear elliptic equations in R^N

We study the quasilinear equation $$(P)-\mathrm{d}\mathrm{i}\mathrm{v}(A(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)+\frac{1}{p}{A}_t(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=g(x,u)\text{in }{\mathbb{R}}^N\text{,}$$ with $N\ge 3$, $p>1$, where $A(x,t)$, $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$and$g(x,t)$areCarath\acute{e}odoryfunctionson${\mathbb R}^N \times {\mathbb R}$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional $\mathcal{J}$ is $C^1$ on the Banach space $X=W^{1,p}({\mathbb{R}}^N)\cap L^{\mathrm{\infty }}({\mathbb{R}}^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of $\mathcal{J}$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_X$ and the norm on $W^{1,p}({\mathbb{R}}^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem.