Radial bounded solutions for modified Schrödinger equations

We study the quasilinear equation (P) − div(a(x, u,∇u)) + At(x, u,∇u) + |u|^(p−2)u = g(x, u) in RN, with N ≥ 2 and p > 1. Here, we suppose A : R^N × R × R^N → R is a given C1-Carathéodory function which grows as |ξ|^p with At(x, t, ξ) = ∂A/∂t (x, t, ξ), a(x, t, ξ) = ∇ξA(x, t, ξ) and g(x, t) is a given Carathéodory function on R^N × R which grows as |ξ|^q with 1 < q < p. Suitable assumptions on A(x, t, ξ) and g(x, t) set off the variational structure of (P) and its related functional J is C1 on the Banach space X = W^(1,p)(R^N) ∩ L^∞(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 Xr, subspace of the radial functions in X. Following an approach which 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.