A Berestycki–lions Theorem Revisited

Zhang, Jianjun; Zou, Wenming

doi:10.1142/s0219199712500332

Cited by 71 publications

(58 citation statements)

References 21 publications

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“…If

v ≢ 0

, then for sufficiently large

R > 0

\underset{ε \to 0}{trueprefixlim inf} \int_{B (y_{ε}, R)} false| \nabla u_{ε} |^{2} \geq \frac{1}{2} \int_{R^{N}} {| \nabla v |}^{2} = \frac{N}{2} L_{V (x_{0})} false(v false) .

Obviously,

L_{V (x_{0})} false(v false) \geq E_{V (x_{0})}

. Now recalling from that

E_{a} > E_{b}

a > b

. Then

\underset{ε \to 0}{trueprefixlim inf} \int_{B (y_{ε}, R)} {| \nabla u_{ε} |}^{2} \geq \frac{N}{2} \underset{1 \leq i \leq k}{trueprefixinf} E_{m_{i}} > 0

which is in contradiction with if d is small enough.…”

Section: Proof Of Theoremsupporting

confidence: 60%

See 1 more Smart Citation

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

Zhang

Yang

2016

Mathematische Nachrichten

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“…If

v ≢ 0

, then for sufficiently large

R > 0

\underset{ε \to 0}{trueprefixlim inf} \int_{B (y_{ε}, R)} false| \nabla u_{ε} |^{2} \geq \frac{1}{2} \int_{R^{N}} {| \nabla v |}^{2} = \frac{N}{2} L_{V (x_{0})} false(v false) .

Obviously,

L_{V (x_{0})} false(v false) \geq E_{V (x_{0})}

. Now recalling from that

E_{a} > E_{b}

a > b

. Then

\underset{ε \to 0}{trueprefixlim inf} \int_{B (y_{ε}, R)} {| \nabla u_{ε} |}^{2} \geq \frac{N}{2} \underset{1 \leq i \leq k}{trueprefixinf} E_{m_{i}} > 0

which is in contradiction with if d is small enough.…”

Section: Proof Of Theoremsupporting

confidence: 60%

“…. Recalling that in [32], the authors proved that, the mountain path level of (2.3) corresponds to the least energy level. Then similar as that in [5], we have…”

Section: Proof Of Theorem 11mentioning

confidence: 98%

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

Zhang

Yang

2016

Mathematische Nachrichten

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“…For small λ > 0, it still remains unknown whether problem (1.3) has a ground state. In [37], we proved that problem (1.3) has a ground state with the assumptions (F 1 )-(F 3 ). Meanwhile, we show that a ground state of (1.3) is a mountain pass solution.…”

Section: Introductionmentioning

confidence: 95%

Standing waves for nonlinear Schrödinger equations involving critical growth

Zhang¹,

Chen²,

Zou³

2014

Journal of the London Mathematical Society

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We consider the following singularly perturbed nonlinear elliptic problem:where N 3 and f is the nonlinearity of critical growth. In this paper, we construct a solution uε of the above problem, which concentrates at an isolated component of the positive local minimum points of V as ε → 0 under certain conditions on f . Our result completes the study made in some very recent works in the sense that, in those papers, only the subcritical growth was considered. T 0 f (t) dt. Let M ≡ {x ∈ O : V (x) = m}.

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“…In the case s = , the hypotheses (H2) ὔ -(H4) ὔ were introduced in Zhang and Zou [40] (see also Alves, Souto and Montenegro [1]) to obtain the ground state of the scalar field equation…”

Section: Where U Is a Radial Groundmentioning

confidence: 99%