Ground state solution for a periodic p&q-Laplacian equation involving critical growth without the Ambrosetti-Rabinowitz condition

We study the ground state solutions for the following p & q-Laplacian equation {-delta(p)u - delta(q)u + V(x)(|u|(p-2)u + |u|(q-2)u) =lambda K(x)f(u) + |u|(q & lowast;-2)u, x is an element of R-N,u is an element of W-1,W-p(RN) & cap; W-1,W-q(R-N), where lambda > 0 is a parameter large enough, delta(r)u = div(|& nabla;u|(r-2)& nabla;u) with r is an element of {p, q} denotes the r-Laplacian operator, 1 < p < q < N and q(& lowast;) = Nq/(N - q). Under some assumptions for the periodic potential V, weight function K and non linearity f without the Ambrosetti-Rabinowitz condition, we show the above equation has a ground state solution.