Abstract: In this paper, we use the concentration-compactness prieciple togther with the Mountain Pass Lemma to get the existence of nontrivial solutions of the following scalar field equations with strong nonlinearity
-∑_i=1^N ∂/∂x_i(|▽u|^p-2 ∂u/∂x_i)+a(x)|u|^q-2 u=f(x,u),x∈R^N,N ≥ 2 u∈E={u∈L^q(R^N)|u real,∂u/∂x_i∈L^p(R^N),1 ≤ i ≤ N}, where 2 ≤ p ≤ q,p=N.