We investigate the problem (equation presented) where ω is a bounded and smooth domain of ℝ N (N ≥ 2), 1 < q < 2 < p, λ > 0, and b ϵ C 1+α (δω) for some α ϵ (0, 1). We show that f δω b < 0 is a necessary and sufficient condition for the existence of nontrivial nonnegative solutions of this problem. Under the additional condition b + ≠ 0 we show that for λ > 0 sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in C(ω) as λ → 0. When p < 2. we also provide the asymptotic profiles of these solutions.
