Existence and Stability of Standing Waves For Schrödinger-Poisson-Slater Equation

Kikuchi, Hiroaki

doi:10.1515/ans-2007-0305

Cited by 75 publications

(66 citation statements)

References 19 publications

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“…Problem with fixed

λ

has been widely studied; see previous studies and the references therein. In those references, it is well known that solutions of are the critical points of the following

{scriptC}^{1}

functional associated with :

I (u) = \frac{1}{2} \int_{R^{3}} | \nabla u |^{2} - \frac{λ}{2} \int_{R^{3}} u^{2} + \frac{κ}{4} \int_{R^{3}} ϕ_{u} u^{2} - \int_{R^{3}} F (u),

where and in the sequel

F false(t false) = \int_{0}^{t} f false(s false) d s

.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Xie

Chen

2019

Math Methods in App Sciences

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In this paper, we study the existence and multiplicity of standing waves with prescribed L2‐norm Schrödinger‐Poisson equations with general nonlinearities in double-struckR3: i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0, where κ>0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional Eκ(u)=12∫R3|∇u|2+κ4∫R3(|x|−1*u2)u2−∫R3F(u) constrained on the L2‐spheres Sfalse(cfalse)={}u∈H1false(double-struckR3false):false|false|ufalse|false|22=c,1emc>0, where Ffalse(sfalse):=∫0sffalse(tfalse)dt. We consider the case where Eκ is unbounded from below on Sfalse(cfalse) and establish the existence of critical points of Eκ on Sfalse(cfalse) for c>0 sufficiently small and under some mild assumptions on f. In addition, we show that there are infinitely many radial critical points false{unκfalse} of Eκ on Sfalse(cfalse) when f is odd and present a convergence property of unκ as κ↘0. Our results generalize some recent results in the literature.

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“…Problem with fixed

λ

has been widely studied; see previous studies and the references therein. In those references, it is well known that solutions of are the critical points of the following

{scriptC}^{1}

functional associated with :

I (u) = \frac{1}{2} \int_{R^{3}} | \nabla u |^{2} - \frac{λ}{2} \int_{R^{3}} u^{2} + \frac{κ}{4} \int_{R^{3}} ϕ_{u} u^{2} - \int_{R^{3}} F (u),

where and in the sequel

F false(t false) = \int_{0}^{t} f false(s false) d s

.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Xie

Chen

2019

Math Methods in App Sciences

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“…In subcritical case, i,e., the nonlinearity g is subcritical growth, there is a greatest part of the literature and we refer the reader to [2,3,9,10,19,22,23,33] and the reference therein. Below we divide the parameter ε into two cases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In [19], the author studied the existence and orbital stability of standing wave solutions for system (1.1), where h is given by…”

Section: Casementioning

confidence: 99%

Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Wang

et al. 2015

Z. Angew. Math. Phys.

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In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter, f is a continuous, superlinear and subcritical nonlinearity, and λ = 0 is a real parameter. Suppose that a(x) has at least one global minimum and b(x) has at least one global maximum. We prove that there are two families of positive solutions for sufficiently small ε > 0, of which one is concentrating on the set of minimal points of a and the other on the sets of maximal points of b. Moreover, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.Mathematics Subject Classification. 35J61 · 35J50 · 35Q55 · 49J40.

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“…In our case, V (x) can be interpreted as a changing pointwise charge distribution. There is a great deal of literature concerning this type of problem in different situations; see for example [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Infinitely many positive solutions for a Schrödinger–Poisson system

d’Avenia

Pomponio

Vaira

2011

Nonlinear Analysis: Theory, Methods & Applications

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Existence and Stability of Standing Waves For Schrödinger-Poisson-Slater Equation

Cited by 75 publications

References 19 publications

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Infinitely many positive solutions for a Schrödinger–Poisson system

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