Generalized quasilinear elliptic equations in R^N

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation −div(a(x, u,∇u)) + At(x, u,∇u) + V (x)|u|p−2u = g(x, u) inR^N with p > 1, N ≥ 2 and V : RN → R suitable measurable positive function. Here, we suppose A : R^N × R × R^N → R is a given C^1-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), V : R^N → R is a suitable measurable function and g : R^N×R → R is a given Carathéodory function which grows as |ξ|^q with 1 < q < p. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t, ξ), V (x) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.