Generalized Quasilinear Elliptic Equations in R^N

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation [GRAPHICS] . with p > 1, N >= 2 and V : R-N. R suitable measurable positive function. Here, we suppose A : R-N x R x R-N -> R is a given C-1 - Caratheodory function which grows as |xi|(p), with A(t)(x, t, xi) = partial derivative A/partial derivative t (x, t, xi), a(x, t, xi) = Delta xi A(x, t, xi), V : R-N -> R is a suitable measurable function and g : R-N xR -> R is a given Carath ' eodory function which grows as |xi|(q) with 1 < q < p. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x, t, xi), 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.