The existence of infinitely many solutions for nonlinear elliptic equations involving pLaplace type operators in R^N

We are concerned with the following nonlinear elliptic equations -div(Φ(x, ∆u))+b(x)│u│^p-2u=λf(x,u) in R^N, where the function Φ(x , v) is of type │v│^p-2v, b : R^N→(0, ∞) is a continuous potential function λ is a real parameter, and f: R^N*R→R is a Carath´eodory function. In this paper, under suitable assumptions, we show the existence of infinitely many weak solutions for the problem above without assuming the Ambrosetti and Rabinowitz condition, by using the fountain theorem. Next, we give a result on the existence of a sequence of solutions for the problem above converging to zero in the L^ ∞- norm by employing the Moser iteration under appropriate conditions.