A singular Yamabe problem on manifolds with solid cones

Proposition A.1 (see [27, Theorem 4.1]).

Suppose n ≥ 2 , a i ⁢ j ∈ L ∞ ⁢ ( B 1 ) and c ∈ L q ⁢ ( B 1 ) for some q > n 2 satisfy

and

for some positive constants λ and Λ. Suppose that u ∈ H 1 ⁢ ( B 1 ) is a subsolution in the sense that the inequality

holds for any ζ ∈ H 0 1 ⁢ ( B 1 ) and ζ ≥ 0 in B 1 . If f ∈ L q ⁢ ( B 1 ) , then u + ∈ L loc ∞ ⁢ ( B 1 ) . Moreover, there holds for any r ∈ ( 0 , 1 ) and p > 0

where C = C ⁢ ( λ , Λ , p , q , n , B 1 ) > 0 .

Lemma A.2.

Suppose f ∈ L q ⁢ ( B ) , f 1 ∈ L q 1 ⁢ ( N ) and

Set

There exist ε = ε ⁢ ( q , q 1 , n ) > 0 and N = N ⁢ ( λ , Λ , n , B , c , c 1 ) > 0 such that if

then

where C = C ⁢ ( λ , Λ , n , B , c , c 1 , q , q 1 ) > 0 .

Proof.

We will follow the proof in [27, Theorem 4.1].

Inequality (A.3) and Claim 1 imply

where v = u h ⁢ ζ 2 . Next, we will estimate the terms on the right-hand side of (A.4).

Let us observe that q > n 2 , q 1 > n - 1 , { u h ζ ≠ 0 } ⊂ 𝚅 ( h , 1 ) , { u h ζ ≠ 0 } ∩ 𝒩 ⊂ 𝚅 𝒩 ( h , 1 ) , | 𝚅 ⁢ ( h , 1 ) | ≤ h - 1 ⁢ ∫ 𝚅 ⁢ ( h , 1 ) u + ⁢ d x , and | 𝚅 𝒩 ⁢ ( h , 1 ) | ≤ h - 1 ⁢ ∫ 𝚅 𝒩 ⁢ ( h , 1 ) u + ⁢ d σ . By (A.4) and Claim 2, there exists a constant N = N ⁢ ( λ , Λ , n , B , c , c 1 ) > 0 such that if h > N ⁢ max ⁡ { ∥ u + ∥ L 2 ⁢ ( B ) , ∥ u + ∥ L 2 ⁢ ( 𝒩 ) , 𝒮 ∗ } , then

and

where C 10 = C 10 ⁢ ( λ , Λ , n , B , c , c 1 ) > 0 .

From (A.5)–(A.7) and (A.9), we have

(A.10)

and

(A.11)

where C i = C i ⁢ ( λ , Λ , n , B , c , c 1 ) > 0 , i = 11 , 12 .

On the other hand. Set ε = 1 n - 1 - 1 min ⁡ { q , q 1 } , by Young’s inequality,

where C 13 = n ⁢ ( 1 + ε ) 2 and C 14 = ( n - 1 ) ⁢ ( 1 + ε ) . Observe that

Inequalities (A.8), (A.10)–(A.13) imply

(A.14)

where C 15 = C 15 ⁢ ( λ , Λ , n , B , c , c 1 , q , q 1 ) > 0 . Consider (A.14) and the following claim, which is proved as in [27]:

Therefore,

where C 16 = C 16 ⁢ ( λ , Λ , n , B , c , c 1 , q , q 1 ) > 0 . This proves Lemma A.2. ∎

Lemma A.3.

Suppose that u ∈ H 1 ⁢ ( B ) , u | D ∈ L ∞ ⁢ ( D ) , S ∗ ≥ ∥ u ∥ L ∞ ⁢ ( D ) , and

for all k ≥ S ∗ and ζ ∈ C c ∞ ⁢ ( K 1 ) with ζ ≥ 0 . Assume f ∈ L q ⁢ ( B ) and f 1 ∈ L q 1 ⁢ ( N ) . If p , p 1 ≥ 2 , then

where C = C ⁢ ( λ , Λ , p , p 1 , q , q 1 , n , B ) > 0 .

Proof.

We again follow the lines of [27, Theorem 4.1]. As observed above, the condition (A.2) is satisfied. Then, by Lemma A.2, we have u + ∈ L ∞ ⁢ ( 2 - 1 ⁢ B ) ∩ L ∞ ⁢ ( 2 - 1 ⁢ 𝒩 ) and

where C = C ⁢ ( λ , Λ , q , q 1 , n , B ) > 0 . Using Hölder’s inequality, we can conclude the proof. ∎

Lemma A.4.

Suppose that u ∈ H 1 ⁢ ( B ) , u | D ∈ L ∞ ⁢ ( D ) , S ∗ ≥ ∥ u ∥ L ∞ , and

for all k ≥ S ∗ , and ζ ∈ C c ∞ ( K 1 ∩ { x n - 1 > 0 } ) and ζ ≥ 0 . Assume f ∈ L q ⁢ ( B ) . If p ≥ 2 and X is a compact set with X ⊂ B ∪ D , then we have

where C = C ⁢ ( λ , Λ , p , q , n , X , B ) > 0 .

Lemma A.5.

Suppose that u ∈ H 1 ⁢ ( B ) and

for all k ≥ 0 and ζ ∈ C c ∞ ( K 1 ∩ { x n > 0 } ) with ζ ≥ 0 . Assume f ∈ L q ⁢ ( B ) and f 1 ∈ L q 1 ⁢ ( N ) . If p , p 1 ≥ 2 and X is a compact set with X ⊂ B ∪ N , then we have

where C = C ⁢ ( λ , Λ , p , p 1 , q , q 1 , n , X , B ) > 0 .