Bounded solutions for quasilinear modified Schrödinger equations

In this paper we establish a new existence result for the quasilinear elliptic problem -div(A(x,u)|∇ u|^p-2∇ u) +1/p A_t(x,u)|∇ u|^p + V(x)|u|^p-2 u = g(x,u) in ℝ^N, with N≥ 2 , p>1 and V:ℝ^N→ℝ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that A:ℝ^N×ℝ→ℝ is a 𝒞^1 -Caratheodory function such that A_t(x,t) = ∂ A/∂ t (x,t) and a given Carathéodory function g:ℝ^N×ℝ→ℝ has a subcritical growth and satisfies the Ambrosetti–Rabinowitz condition. Since the coefficient of the principal part depends also on the solution itself, we study the interaction of two different norms in a suitable Banach space so to obtain a “good” variational approach. Thus, by means of approximation arguments on bounded sets we can state the existence of a nontrivial weak bounded solution.