Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement

Bellazzini, Jacopo; Boussaïd, Nabile; Jeanjean, Louis; Visciglia, Nicola

doi:10.1007/s00220-017-2866-1

Cited by 94 publications

(143 citation statements)

References 35 publications

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“…To overcome this difficulty, for any r > 0, we prove that there exists c 0 , which is depending on r and p , such that for 0 < c ≤ c 0 ,

\inf_{u \in S false(c false) \cap B false(\frac{r}{4} false)} I false(u false) < \inf_{u \in S false(c false) \cap false(B false(r false) \ B false(\frac{r}{2} false) false)} I false(u false),

then the minimizer of

m_{c}^{r}

stays away from the boundary of S ( c ) ∩ B ( r ) and hence is indeed a critical point of I ( u ) constrained on S ( c ). We point out here that, in Bellazzini et al, a similar inequality as was proved, but the sets are replaced by

S false(c false) \cap B false(\frac{c r}{2} false)

and S ( c ) ∩ ( B ( r )\ B ( cr )), which leads to a necessary condition r ≥ 2Λ 0 , where

Λ_{0} = \inf false(s p e c false(- true \sum_{i = 1}^{3} \partial_{x_{i}}^{2} + false(x_{1}^{2} + x_{2}^{2} false) false) false)

. In fact, the existence of local minimum obtained in Bellazzini et al requires that r ≥ 2Λ 0 (although it seems to be ignored in Bellazzini et al).…”

Section: Introduction and Main Resultsmentioning

confidence: 51%

“…We point out here that, in Bellazzini et al, a similar inequality as was proved, but the sets are replaced by

S false(c false) \cap B false(\frac{c r}{2} false)

and S ( c ) ∩ ( B ( r )\ B ( cr )), which leads to a necessary condition r ≥ 2Λ 0 , where

Λ_{0} = \inf false(s p e c false(- true \sum_{i = 1}^{3} \partial_{x_{i}}^{2} + false(x_{1}^{2} + x_{2}^{2} false) false) false)

. In fact, the existence of local minimum obtained in Bellazzini et al requires that r ≥ 2Λ 0 (although it seems to be ignored in Bellazzini et al). However, according to our method, the local minimum exists for all r > 0 and 0 < c ≤ c 0 .…”

Section: Introduction and Main Resultsmentioning

confidence: 51%

“…Recently, Bellazzini et al studied the existence of normalized solutions for superlinear NLS with a partial confinement

- Δ u + (x_{1}^{2} + x_{2}^{2}) u - | u {false|}^{p - 2} u = λ u, x \in {double-struckR}^{3},

where

\frac{10}{3} \leq p < 6

, and showed the problem has a critical point v with

\int_{R^{3}} false| v |^{2} = c

, which is a local minimum.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Motivated by Bellazzini et al, we try to prove that I ( u ) has at least two critical points constrained on S ( c ). To state our main results, we define the following:

‖ u {false‖}_{\dot{normalΣ}} : = {(\int_{{double-struckR}^{3}} | \nabla u {false|}^{2} + \int_{{double-struckR}^{3}} | x {false|}^{2} u^{2})}^{\frac{1}{2}},

and

B (r) = {u \in Σ | ‖ u ‖_{\dot{normalΣ}}^{2} \leq r, r > 0} .

We consider the following local minimization problem

m_{c}^{r} : = \inf_{u \in S false(c false) \cap B false(r false)} I false(u false) .

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where 10 3 ≤ p < 6, and showed the problem has a critical point v with ∫ R 3 |v| 2 = c, which is a local minimum. Motivated by Bellazzini et al, 10 we try to prove that I(u) has at least two critical points constrained on S(c). To state our main results, we define the following:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Multiplicity and stability of standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential

Luo

2019

Math Methods in App Sciences

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“…To overcome this difficulty, for any r > 0, we prove that there exists c 0 , which is depending on r and p , such that for 0 < c ≤ c 0 ,

\inf_{u \in S false(c false) \cap B false(\frac{r}{4} false)} I false(u false) < \inf_{u \in S false(c false) \cap false(B false(r false) \ B false(\frac{r}{2} false) false)} I false(u false),

then the minimizer of

m_{c}^{r}

S false(c false) \cap B false(\frac{c r}{2} false)

and S ( c ) ∩ ( B ( r )\ B ( cr )), which leads to a necessary condition r ≥ 2Λ 0 , where

Λ_{0} = \inf false(s p e c false(- true \sum_{i = 1}^{3} \partial_{x_{i}}^{2} + false(x_{1}^{2} + x_{2}^{2} false) false) false)

. In fact, the existence of local minimum obtained in Bellazzini et al requires that r ≥ 2Λ 0 (although it seems to be ignored in Bellazzini et al).…”

Section: Introduction and Main Resultsmentioning

confidence: 51%

“…We point out here that, in Bellazzini et al, a similar inequality as was proved, but the sets are replaced by

S false(c false) \cap B false(\frac{c r}{2} false)

and S ( c ) ∩ ( B ( r )\ B ( cr )), which leads to a necessary condition r ≥ 2Λ 0 , where

Λ_{0} = \inf false(s p e c false(- true \sum_{i = 1}^{3} \partial_{x_{i}}^{2} + false(x_{1}^{2} + x_{2}^{2} false) false) false)

Section: Introduction and Main Resultsmentioning

confidence: 51%

“…Recently, Bellazzini et al studied the existence of normalized solutions for superlinear NLS with a partial confinement

- Δ u + (x_{1}^{2} + x_{2}^{2}) u - | u {false|}^{p - 2} u = λ u, x \in {double-struckR}^{3},

where

\frac{10}{3} \leq p < 6

, and showed the problem has a critical point v with

\int_{R^{3}} false| v |^{2} = c

, which is a local minimum.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Motivated by Bellazzini et al, we try to prove that I ( u ) has at least two critical points constrained on S ( c ). To state our main results, we define the following: