Embeddings of weighted Sobolev spaces and degenerate elliptic problems

Zhang, Guo; Mei, Linfeng; Wan, Fangshu; Guan, Xiaohong

doi:10.1007/s11425-016-0403-6

Cited by 8 publications

(18 citation statements)

References 25 publications

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“…We argue by contradiction. Assume that there exists a sequence

false{φ_{m} false} \subset C_{0}^{\infty} false(normalΩ false)

such that

0true \int_{Ω} b (x) {false| φ_{m} false|}^{p_{s} + 1} d x = 1, for all m \in boldR,

0true \int_{Ω} a (x) {false| \nabla φ_{m} false|}^{2} d x \to 0 as m \to \infty .

By the same arguments as those in the proof of Lemma 2.1 of , we obtain that, up to a subsequence of

{φ_{m}}

φ_{m} \to 0 in H_{l o c}^{1} ({boldR}^{N} ∖ Z true) as m \to \infty,

where and in the following

Z = {z_{1}, z_{2}, \dots, z_{k}}

. Now we fix an index

i \in {1, 2, \dots, k}

, it is easily seen from and the standard embedding that (up to a subsequence of

{φ_{m}}

)

φ_{m} \to 0 in H^{\frac{1}{2}} false(\partial B_{i} false)

…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 90%

“…In a recent paper , the authors established embeddings of weighted Sobolev spaces with the weights

a (x)

and

b (x)

satisfying –, but there is a crucial assumption for each

(θ_{i}, l_{i})

θ_{i} \geq \frac{(N - 2)}{2} τ_{i} .

We see that

1 < p_{s}^{i} \leq \frac{N + 2}{N - 2}

for any

1 \leq i \leq k

provided the assumption (CA) holds.…”

Section: Introductionmentioning

confidence: 94%

“…Proof The proof is a variant of the proof of Lemma 2.1 of . We still need to explain some details here, since unlike to Lemma 2.1 of , the assumption (CA) may not hold for some

i \in {1, 2, \dots, k}

.…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 99%

“…Now we fix an index

i \in {1, 2, \dots, k}

, it is easily seen from and the standard embedding that (up to a subsequence of

{φ_{m}}

)

φ_{m} \to 0 in H^{\frac{1}{2}} false(\partial B_{i} false) as m \to \infty .

We now consider two cases: (i)

θ_{i} \geq 0

, (ii)

θ_{i} < 0

. For the case (i), we denote

φ_{m}^{i}

to be the unique solution of the problem

({, \begin{matrix} - Δ φ_{m}^{i} = 0, & in B_{i}, \\ φ_{m}^{i} = φ_{m}, & on \partial B_{i} . \end{matrix})

By the same arguments as those in the proof of Lemma 2.1 of , we have

φ_{m}^{i} \to 0 strongly in H^{1} false(B_{i} false) as m \to \infty,

and

0true \int_{B_{i}} {false| x - z_{i} false|}^{θ_{i}} ([] {(φ_{m}^{i})}^{2} + {false| \nabla φ_{m}^{i} false|}^{2}) d x

…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 99%

“…Remark Proposition is a new embedding result. Proposition cannot be obtained directly from since under the assumptions of Proposition , (CA) may not hold for some

i \in {1, 2, \dots, k}

, but we need that (CA) hold for all

i \in {1, 2, \dots, k}

in . This implies that, in , we can use the Caffarelli–Kohn–Nirenberg inequality in any

B_{i}

, but we need to use the inequality to replace the Caffarelli–Kohn–Nirenberg inequality in some

B_{i}

here.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Existence and regularity of positive solutions of a degenerate elliptic problem

Zhang

Guan

Wan

2018

Mathematische Nachrichten

Self Cite

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Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form −divfalse(a(x)∇ufalse)=bfalse(xfalse)up in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N≥1, are obtained via new embeddings of some weighted Sobolev spaces with singular weights a(x) and b(x). It is seen that a(x) and b(x) admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.

show abstract

“…We argue by contradiction. Assume that there exists a sequence

false{φ_{m} false} \subset C_{0}^{\infty} false(normalΩ false)

such that

0true \int_{Ω} b (x) {false| φ_{m} false|}^{p_{s} + 1} d x = 1, for all m \in boldR,

0true \int_{Ω} a (x) {false| \nabla φ_{m} false|}^{2} d x \to 0 as m \to \infty .

By the same arguments as those in the proof of Lemma 2.1 of , we obtain that, up to a subsequence of

{φ_{m}}

φ_{m} \to 0 in H_{l o c}^{1} ({boldR}^{N} ∖ Z true) as m \to \infty,

where and in the following

Z = {z_{1}, z_{2}, \dots, z_{k}}

. Now we fix an index

i \in {1, 2, \dots, k}

, it is easily seen from and the standard embedding that (up to a subsequence of

{φ_{m}}

)

φ_{m} \to 0 in H^{\frac{1}{2}} false(\partial B_{i} false)

…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 90%

“…In a recent paper , the authors established embeddings of weighted Sobolev spaces with the weights

a (x)

and

b (x)

satisfying –, but there is a crucial assumption for each

(θ_{i}, l_{i})

θ_{i} \geq \frac{(N - 2)}{2} τ_{i} .

We see that

1 < p_{s}^{i} \leq \frac{N + 2}{N - 2}

for any

1 \leq i \leq k

provided the assumption (CA) holds.…”

Section: Introductionmentioning

confidence: 94%

“…Proof The proof is a variant of the proof of Lemma 2.1 of . We still need to explain some details here, since unlike to Lemma 2.1 of , the assumption (CA) may not hold for some

i \in {1, 2, \dots, k}

.…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 99%

“…Now we fix an index

i \in {1, 2, \dots, k}

, it is easily seen from and the standard embedding that (up to a subsequence of

{φ_{m}}

)

φ_{m} \to 0 in H^{\frac{1}{2}} false(\partial B_{i} false) as m \to \infty .

We now consider two cases: (i)

θ_{i} \geq 0

, (ii)

θ_{i} < 0

. For the case (i), we denote

φ_{m}^{i}

to be the unique solution of the problem

({, \begin{matrix} - Δ φ_{m}^{i} = 0, & in B_{i}, \\ φ_{m}^{i} = φ_{m}, & on \partial B_{i} . \end{matrix})

By the same arguments as those in the proof of Lemma 2.1 of , we have

φ_{m}^{i} \to 0 strongly in H^{1} false(B_{i} false) as m \to \infty,

and

0true \int_{B_{i}} {false| x - z_{i} false|}^{θ_{i}} ([] {(φ_{m}^{i})}^{2} + {false| \nabla φ_{m}^{i} false|}^{2}) d x

…”

Section: Sobolev Type Embeddings: Proof Of Propositions Andmentioning

confidence: 99%

“…Remark Proposition is a new embedding result. Proposition cannot be obtained directly from since under the assumptions of Proposition , (CA) may not hold for some