Assume that the conclusion does not hold, then there exist a sequence in with such that.
Therefore ({u^_{varepsilon}}) is non-vanishing, that is, there exist a sequence ({x_{varepsilon}}subset mathbb{R}^{N}) and constant (R>0), (sigma>0) such that (lim_{varepsilonrightarrow0}int_{B_{R}(x_{varepsilon})}u^{*2}_{varepsilon}geqsigma).
Lemma 2.8 If { u n } is bounded in H and u n does not converge to 0 in measure, then there exist a sequence { x n k } ⊂ Z and a subsequence { u n k } of { u n } such that u n k ( ⋅ + x n k T ) ⇀ u ≠ 0 in H 1 ( R ).
Moreover, if { u n } is a minimizing sequence for the problem ( I q ∗ ), then there exist a sequence { y n } ⊂ R and g ∈ G q ∗ such that { u n ( ⋅ + y n ) } contains a subsequence converging strongly in H 1 ( R ) to g, and lim n → + ∞ inf g ∈ G q ∗ ∥ u n − g ∥ = 0, where ∥ ⋅ ∥ is the norm of H 1 ( R ).
Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exist a sequence { y n } ⊂ R and g ∈ G q such that { u n ( ⋅ + y n ) } contains a subsequence converging strongly in H 1 ( R ) to g, and lim n → + ∞ inf g ∈ G q ∥ u n − g ∥ = 0. Theorem 3.2 Let α = 1 and f ( u ) = 1 p u p, where 1 < p < 5.
Recall that S is said to be an asymptotically strict pseudocontraction iff there exist a sequence { k n } ⊂ [ 1, ∞ ) with k n → 1 as n → ∞ and a constant κ ∈ [ 0, 1 ) such that ∥ S n x − S n y ∥ 2 ≤ k n ∥ x − y ∥ 2 + κ ∥ ( I − S n ) x − ( I − S n ) y ∥ 2, ∀ x, y ∈ C, n ≥ 1. For such a case, S is also said to be an asymptotically κ-strict pseudocontraction.
Since, there exist a sequence satisfying.
If there exist a sequence in and a real number such that for every, (3.1).
Assume that there exist a sequence and constants such that (6.10).
From (2), there exist a sequence in and a point such that.
Then by Lemma 2.1, for, there exist a sequence of functions, a sequence of complex number, and such that (3.9).
