On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in RN

In this paper, we consider the attainability of a maximizing problem D : = sup ‖ u ‖ H γ 1 , N = 1 ‖ u ‖ N N + α ‖ u ‖ p p , where N ≥ 2 , N < p < ∞ , α > 0 , 0 < γ ≤ N and ‖ u ‖ H γ 1 , N = ‖ u ‖ N γ + ‖ ∇ u ‖ N γ 1 γ . The existence of a maximizer for D is closely related to the exponent γ . In fact, we show that the value α = α ∗ : = inf ‖ u ‖ H γ 1 , N = 1 1 - ‖ u ‖ N N ‖ u ‖ p p is a threshold in terms of the attainability of D .