Radial bounded solutions for modified Schrödinger equations

We study the quasilinear equation (P)\qquad - {\rm div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in\R^N,} with $N\ge 3$ and $p>1$. Here, we suppose $A:{\mathbb{R}}^N\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is a given $C^1$-Carathéodory function which grows as $\mathrm{\mid }\xi {\mathrm{\mid }}^p$ with $A_t(x,t,\xi )=\frac{\mathrm{\partial }A}{\mathrm{\partial }t}(x,t,\xi )$, $a(x,t,\xi )={\mathrm{\nabla }}_{\xi }A(x,t,\xi )$ and $g(x,t)$ is a given Carathéodory function on ${\mathbb{R}}^N\times \mathbb{R}$ which grows as $\mathrm{\mid }\xi {\mathrm{\mid }}^q$ with $1<q<p$. Suitable assumptions on $A(x,t,\xi )$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional $\text{J}$ 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 $\text{J}$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach that exploits the interaction between the intersection norm in $X$ and the norm on $W^{1,p}({\mathbb{R}}^N)$, we prove the existence of at least two weak bounded radial solutions of $(P)$, one positive and one negative, by applying a generalized version of the Minimum Principle.