Radial bounded solutions for modified Schr\"odinger 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 $A : \R^N \times \R \times \R^N \to \R$ is a given ${C}^{1}$-Carath\'eodory function which grows as $|\xi|^p$ with $A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)$, $a(x,t,\xi) = \nabla_\xi A(x,t,\xi)$ and $g(x,t)$ is a given Carath\'eodory function on $\R^N \times \R$ which grows as $|\xi|^q$ 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 $\J$ is $C^1$ on the Banach space $X = W^{1,p}(\R^N) \cap L^\infty(\R^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of $\J$ 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 $W^{1,p}(\R^N)$, 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.