Standing waves for nonlinear Schrödinger equations involving critical growth

Zhang, Jianjun; Chen, Zhijie; Zou, Wenming

doi:10.1112/jlms/jdu054

Cited by 53 publications

(35 citation statements)

References 35 publications

(69 reference statements)

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“…If

v \equiv 0

, i.e.,

v_{ε} ⇀ 0

weakly in

H^{1} (R^{N})

, then

v_{ε} \to 0

strongly in

L_{l o c}^{q} (R^{N})

for

q \in [2, 2^{*})

. Thus, by Sobolev's embedding theorem, there exists

C > 0

(independent of ε) such that, for ε small,

\int_{B (0, 1)} {| \nabla v_{ε} |}^{2} \geq C r^{\frac{2}{2^{*}}} > 0 .

Similar as that in , we can prove that

\underset{ε \to 0}{trueprefixlim} \underset{\overset{ϕ \in C_{0}^{\infty} (Ω)}{false∥ ϕ false∥ = 1}}{trueprefixsup} false| ⟨ ρ_{ε}, ϕ ⟩ false| = 0,

where

Ω = B false(0, 2 false), ρ_{ε} \in H^{- 1} (R^{N})

and

ρ_{ε} = Δ v_{ε} + {| v_{ε} |}^{2^{*} - 2} v_{ε}

. It follows from Lemma that there exists

{\overset{y}{true}}_{ε} \in {double-struckR}^{N}

and

σ_{ε}

…”

Section: Proof Of Theoremsupporting

confidence: 56%

“…By , it is easy to see

u_{ε}^{2} \to 0

strongly in

H_{ε}

and thus the conclusion follows directly. Finally, we give the sketch of the proof of the Claim similarly as that in . By Lemma , assume by contradiction that there exists

r > 0

, such that for some

i = 1, 2, \dots, k

\underset{ε \to 0}{trueprefixlim inf} \underset{z \in {double-struckR}^{N}}{trueprefixsup} \int_{B (z, 1)} {(|| w_{ε}^{i} - w^{i})}^{2^{*}} = 2 r > 0 .

Then, there exists

z_{ε} \in {double-struckR}^{N}

such that

{trueprefixlim inf}_{ε \to 0} \int_{B (z_{ε}, 1)} | w_{ε}^{i} - w^{i} {false|}^{2^{*}} > r

.…”

Section: Proof Of Theoremmentioning

confidence: 66%

“…Then for ε small,

\int_{B (0, a + 1)} {| v_{ε} |}^{2^{*}} > r,

where

v_{ε} = w_{ε}^{i} - w^{i}

and

v_{ε} ⇀ 0

weakly in

H^{1} (R^{N})

. Similar as that in ,

\underset{ε \to 0}{trueprefixlim} \underset{\overset{ϕ \in C_{0}^{\infty} (Ω)}{false∥ ϕ false∥ = 1}}{trueprefixsup} false| ⟨ ρ_{ε}, ϕ ⟩ false| = 0,

where

Ω = B false(0, a + 2 false), ρ_{ε} \in H^{- 1} (R^{N})

and

ρ_{ε} = Δ v_{ε} + {| v_{ε} |}^{2^{*} - 2} v_{ε}

. It follows from Lemma that there exist

{\overset{z}{true}}_{ε} \in {double-struckR}^{N}

and

σ_{ε} > 0

, such that

{\overset{z}{true}}_{ε} \to \tilde{z} \in \bar{B (0, a + 1)}

σ_{ε} \to 0

and

{\overset{w}{true}}_{...}

…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…Proof The proof is similar to [, Proposition ], here we only give the sketch. For convenience, we write ε for

ε_{j}

.…”

Section: Proof Of Theoremmentioning

confidence: 96%

See 3 more Smart Citations

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

Zhang

Yang

2016

Mathematische Nachrichten

Self Cite

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show abstract

“…If

v \equiv 0

, i.e.,

v_{ε} ⇀ 0

weakly in

H^{1} (R^{N})

, then

v_{ε} \to 0

strongly in

L_{l o c}^{q} (R^{N})

for

q \in [2, 2^{*})

. Thus, by Sobolev's embedding theorem, there exists

C > 0

(independent of ε) such that, for ε small,

\int_{B (0, 1)} {| \nabla v_{ε} |}^{2} \geq C r^{\frac{2}{2^{*}}} > 0 .

Similar as that in , we can prove that

\underset{ε \to 0}{trueprefixlim} \underset{\overset{ϕ \in C_{0}^{\infty} (Ω)}{false∥ ϕ false∥ = 1}}{trueprefixsup} false| ⟨ ρ_{ε}, ϕ ⟩ false| = 0,

where

Ω = B false(0, 2 false), ρ_{ε} \in H^{- 1} (R^{N})

and

ρ_{ε} = Δ v_{ε} + {| v_{ε} |}^{2^{*} - 2} v_{ε}

. It follows from Lemma that there exists

{\overset{y}{true}}_{ε} \in {double-struckR}^{N}

and

σ_{ε}

…”

Section: Proof Of Theoremsupporting

confidence: 56%

“…By , it is easy to see

u_{ε}^{2} \to 0

strongly in

H_{ε}

and thus the conclusion follows directly. Finally, we give the sketch of the proof of the Claim similarly as that in . By Lemma , assume by contradiction that there exists

r > 0

, such that for some

i = 1, 2, \dots, k

\underset{ε \to 0}{trueprefixlim inf} \underset{z \in {double-struckR}^{N}}{trueprefixsup} \int_{B (z, 1)} {(|| w_{ε}^{i} - w^{i})}^{2^{*}} = 2 r > 0 .

Then, there exists

z_{ε} \in {double-struckR}^{N}

such that

{trueprefixlim inf}_{ε \to 0} \int_{B (z_{ε}, 1)} | w_{ε}^{i} - w^{i} {false|}^{2^{*}} > r

.…”

Section: Proof Of Theoremmentioning

confidence: 66%

“…Then for ε small,

\int_{B (0, a + 1)} {| v_{ε} |}^{2^{*}} > r,

where

v_{ε} = w_{ε}^{i} - w^{i}

and

v_{ε} ⇀ 0

weakly in

H^{1} (R^{N})

. Similar as that in ,

\underset{ε \to 0}{trueprefixlim} \underset{\overset{ϕ \in C_{0}^{\infty} (Ω)}{false∥ ϕ false∥ = 1}}{trueprefixsup} false| ⟨ ρ_{ε}, ϕ ⟩ false| = 0,

where

Ω = B false(0, a + 2 false), ρ_{ε} \in H^{- 1} (R^{N})

and

ρ_{ε} = Δ v_{ε} + {| v_{ε} |}^{2^{*} - 2} v_{ε}

. It follows from Lemma that there exist

{\overset{z}{true}}_{ε} \in {double-struckR}^{N}

and

σ_{ε} > 0

, such that

{\overset{z}{true}}_{ε} \to \tilde{z} \in \bar{B (0, a + 1)}

σ_{ε} \to 0

and

{\overset{w}{true}}_{...}

…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…Proof The proof is similar to [, Proposition ], here we only give the sketch. For convenience, we write ε for

ε_{j}

.…”

Section: Proof Of Theoremmentioning

confidence: 96%

See 2 more Smart Citations

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

Zhang

Yang

2016

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are some results under stronger assumptions on V (x). We firstly refer to the work by Benci and Cerami [5], they considered the following problem: is sufficiently small, Chabrowski and Yang [8] proved that the problem (1.4) admits cat{ } many solutions; We also refer the readers to paper by Zhang, Chen and Zou [22] for the critical growth problems. Recently, Tang [21] considered the problem (1.1) with critical exponents and indefinite potential function, i.e., a(x) ≥ 0, λ > 0 is a parameter and δ > 0 is a constant which can be arbitrary large such that the operator − + λa(x) − δ is indefinite, under some suitable assumptions on a(x) and δ, the author proved the existence of the least energy solution which localized near the potential well inta −1 (0) for λ large.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells

Guo

Tang

2015

Journal of Differential Equations

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In this paper, we consider the following Schrödinger equations with critical growthwhere N ≥ 4, 2 * is the critical Sobolev exponent, a(x) ≥ 0 and its zero sets are not empty, λ > 0 is a parameter, δ > 0 is a constant such that the operator (− + λa(x) − δ) might be indefinite for λ large. We prove that if the zero sets of a(x) have several isolated connected components 1 , · · · , k such that the interior of i (i = 1, 2, . . . , k) is not empty and ∂ i (i = 1, 2, . . . , k) is smooth. Then for λ sufficiently large, the equation admits, for any i ∈ {1, 2, · · · , k}, a solution which is trapped in a neighborhood of i . The key ingredients of the paper are using a flow argument and a combination of global linking and local linking.

show abstract

Single‐peak solutions for a subcritical Schrödinger equation with non‐power nonlinearity

Liu

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

This paper deals with the following slightly subcritical Schrödinger equation: where is a nonnegative smooth function, , , , . Most of the previous works for the Schrödinger equations were mainly investigated for power‐type nonlinearity. In this paper, we will study the case when the nonlinearity is a non‐power nonlinearity. We show that, for ε small enough, there exists a family of single‐peak solutions concentrating at the positive stable critical point of the potential .

show abstract

Standing waves for nonlinear Schrödinger equations involving critical growth

Cited by 53 publications

References 35 publications

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells

Single‐peak solutions for a subcritical Schrödinger equation with non‐power nonlinearity

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