Quasilinear elliptic problem without Ambrosetti-Rabinowitz condition involving a potential in Musielak-Sobolev spaces setting

In this paper, we consider the quasilinear elliptic problem with potential (p){-div(phi(x, vertical bar del u vertical bar del u) + V(x)vertical bar u vertical bar(q(x)-2) u = f(x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N (N >= 2), V is a given function in a generalized Lebesgue space L-s(x) (Omega), and f(x,u) is a Caratheodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for (P) in the framework of Musielak-Sobolev spaces. The main difficulty here is that the nonlinearity f(x,u) considered does not satisfy the well-known Ambrosetti-Rabinowitz condition.