Abstract: In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem
-Σ_i=1^N∂/∂x_i(|▽u|^p-2∂u/∂x_i)+a(x)|u|^q-2u=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,if N ≤ p;2 ≤ p ≤ q < Np/(N-p),if N > p.
where u|→∫_R^N F(x,u)dx(F(x,t)=∫₀^t f(x,s)ds) need not be compact operators from E to R¹.