where a > 0, λ < 0 and f has an exponential critical growth when N = 2, and f(t) = μ|u|^q−2u+|u|^2∗−2u with q∈(2+4/N ,2^∗), μ > 0 and 2^*=2N/N−2 when N ≥ 3. Our main results complement some recent results for N≥3 and it is totally new for N=2. This is a joint work with Olimpio Miyagaki and Chao Ji.