Existence of multi-bump solutions for the fractional Schrödinger-Poisson system

Liu, Weiming

doi:10.1063/1.4963172

Cited by 21 publications

(11 citation statements)

References 28 publications

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“…under some appropriate conditions on K(x), Q(x) and f ∈ C 1 (R 3 ) behaving like |u| p−2 u with 4 < p < 2 * s = 6 3−2s , where the existence and concentration of positive ground state solutions were obtained. Other interesting results on fractional Schrödinger-Poisson system can be found in [28,29,37,40,42,46] and their references.…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

Chen

Tang

Peng

2018

Studia Scientiarum Mathematicarum Hungarica

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The present study is concerned with the following fractional Schrödinger-Poisson system with steep potential well:where s, t ∈ (0, 1) with 4s + 2t > 3, and λ > 0 is a parameter. Under certain assumptions on V(x), K(x) and f (u) behaving like |u| q−2 u with 2 < q < 2 * s = 6 3−2s , the existence of positive ground state solutions and concentration results are obtained via some new analytical skills and Nehair-Pohožaev identity. In particular, the monotonicity assumption on the nonlinearity is not necessary.

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Section: Introduction and Main Resultsmentioning

confidence: 91%

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

Chen

Tang

Peng

2018

Studia Scientiarum Mathematicarum Hungarica

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“…However, the research of the fractional Schördinger‐Poisson system is not so fruitful (please see other works for example). Meanwhile, to the best knowledge of us, there are few results on the Schrödinger equation involving a Bessel operator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To know more about the study of fractional Schrödinger equations, the reader can refer to related works 2,3,10-20 for example. However, the research of the fractional Schördinger-Poisson system is not so fruitful (please see other works [21][22][23][24] for example). Meanwhile, to the best knowledge of us, there are few results on the Schrödinger equation involving a Bessel operator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and concentration results for fractional Schrödinger‐Poisson systems involving a Bessel operator

Shen

2018

Math Methods in App Sciences

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This paper is concerned with the fractional Schrödinger‐Poisson systems involving a Bessel operator. By using Mountain‐pass theorem and Ekeland's variational principle, we obtain the multiplicity and concentration of nontrivial solutions for the given problem. In particular, although there exist concave‐convex nonlinearities in our problem, it is not necessary to assume that the corresponding Lebesgue norm of the weight function of the convex term needs to be small enough.

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“…Using the monotonicity trick and global compactness Lemma, Teng proved the existence of a nontrivial ground state solution for the system

\{\begin{array}{l} {false(- normalΔ false)}^{s} u + V false(x false) u + ϕ false(x false) u = μ {(||, u)}^{q - 1} u + {(||, u)}^{2_{s}^{*} - 2} u, x \in R^{3}, \\ {false(- normalΔ false)}^{t} ϕ = u^{2}, x \in R^{3}, \end{array}

where

μ \geq 0, 1 < q < 2_{s}^{*} - 1, s, t \in false(0, 1 false), 2 t + 2 s > 3

. By the perturbation method, Li studied the above system for the case V ( x )=1, see previous studies and so on. To the best our knowledge, there is little information on the existence of solution for fractional Schrödinger‐Poisson system in the case

4 < p, q \leq 2_{s}^{*}

.…”

Section: Introductionmentioning

confidence: 99%

Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities

Zi-an

2019

Math Methods in App Sciences

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In this paper, we are concerned with a class of fractional Schrödinger-Poisson system involving subcritical or critical nonlinearities. By using the Nehari manifold and variational methods, we obtain the existence and multiplicity of nontrivial solutions. KEYWORDS critical Sobolev exponents, fractional Schrödinger-Poisson system, Nehari manifold, variational methods MSC CLASSIFICATION 35A15; 35J60; 35R11; 35R09 { −Δu + V(x)u + (x)u = (x, u), x ∈ R 3 , −Δ = u 2 , x ∈ R 3 .

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Existence of multi-bump solutions for the fractional Schrödinger-Poisson system

Abstract: This paper considers the fractional Schrödinger-Poisson system in ℝ3. We prove that the problem has m-bump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multi-bump solutions as ϵ → 0.

Cited by 21 publications

References 28 publications

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

Multiplicity and concentration results for fractional Schrödinger‐Poisson systems involving a Bessel operator

Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities

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