Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system

Sun, Juntao; Wu, Tsung‐fang; Feng, Zhaosheng

doi:10.1016/j.jde.2015.09.002

Cited by 71 publications

(40 citation statements)

References 30 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%

“…In such a case, the frequency cannot be fixed any more, and it appears as a Lagrange parameter with respect to the mass constraint. Problem (2) with fixed has been widely studied; see previous studies [10][11][12][13][14][15][16][17][18] and the references therein. In those references, it is well known that solutions of (2) are the critical points of the following  1 functional associated with (2):…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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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%

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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“…The corresponding results were further generalized by Zhao LG and Zhao FK by considering the cases

p \in false(3, 6 false)

. Recently, Sun, Wu, and Feng used the argument of the filtration of Nehari manifold to show the presence of multiple positive solutions for

2 < p < 6

and ground state solutions for

\frac{1 + \sqrt{73}}{3} < p \leq 4

with weight functions

V

and

K

. We refer to previous studies for more related results for .…”

Section: Introductionmentioning

confidence: 96%

Multibump solutions for nonlinear Schrödinger‐Poisson systems

Chen

2020

Math Meth Appl Sci

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In this paper, we study the following Schrödinger‐Poisson equations: −ε2normalΔu+Vfalse(xfalse)u+Kfalse(xfalse)ϕu=false|ufalse|p−2u,x∈double-struckR3,−ε2normalΔϕ=Kfalse(xfalse)u2,x∈double-struckR3, where p∈false(4,6false), ε>0 is a parameter and V and K satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.

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“…Recently, Schrödinger-Poisson systems on unbounded domains or on the whole space R N have attracted a lot of attention. Many solvability conditions on the nonlinearity have been given to obtain the existence and multiplicity of solutions for Schrödinger-Poisson systems in R N , we refer the readers to [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Compared with the whole space case, there are few works concerning the Schrödinger-Poisson system on a bounded domain; see, for instance, [20,[24][25][26][27][28][29][30][31]. Ruiz and Siciliano [26] studied the following system:…”

Section: Introductionmentioning

confidence: 99%

Existence and multiplicity of nontrivial solutions for Schrödinger-Poisson systems on bounded domains

2018

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In this paper, we concern with the following Schrödinger-Poisson system:where is a smooth bounded domain in R 3 . Under more appropriate assumptions on f , we obtain new results on the existence of nontrivial solutions and infinitely many solutions by using the mountain pass theorem and the symmetric mountain pass theorem, respectively. We extend and improve some recent results in the literature. MSC: 35J20; 35J60

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Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system

Cited by 71 publications

References 30 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

Multibump solutions for nonlinear Schrödinger‐Poisson systems

Existence and multiplicity of nontrivial solutions for Schrödinger-Poisson systems on bounded domains

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