Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent

Teng, Kaimin

doi:10.1016/j.jde.2016.05.022

Cited by 153 publications

(87 citation statements)

References 65 publications

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“…A fractional Schrödinger–Poisson system with

V = 0

and a general nonlinearity in the subcritical and critical case was considered in , where a positive solution was obtained by using a perturbation approach, and the asymptotic behavior of solutions for a vanishing parameter was also given. In and , Teng adapted the monotonicity trick (see e.g. ) to obtain the existence of ground state solutions to the critical and subcritical cases, respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof The proof is similar to the proof of Lemma 3.3 in . For the sake of completeness, we give the details here.…”

Section: The Autonomous Problemmentioning

confidence: 86%

“…The properties of the function

ϕ_{u}^{t}

are given in the following lemma (see [, Lemma 2.3]). Lemma If

4 s + 2 t \geq 3

, then for any

u \in H^{s} ({double-struckR}^{3})

, we have (i)

ϕ_{u}^{t} \geq 0

; (ii)

ϕ_{u}^{t} : H^{s} ({double-struckR}^{3}) \to {scriptD}^{t, 2} ({double-struckR}^{3})

is continuous and maps bounded sets into bounded sets; (iii)

\int_{R^{3}} ϕ_{u}^{t} u^{2} d x \leq {C ∥ u ∥}_{\frac{12}{3 + 2 t}}^{4} \leq C {∥ u ∥}^{4}

; (iv)If

u_{n} ⇀ u

H^{s} ({double-struckR}^{3})

, then

ϕ_{u_{n}}^{t} ⇀ ϕ_{u}^{t}

{scriptD}^{t, 2} ({double-struckR}^{3})

; (v)If

u_{n} \to u

H^{s} ({double-struckR}^{3})

, then

ϕ_{u_{n}}^{t} \to ϕ_{u}^{t}

{scriptD}^{t, 2} ({double-struckR}^{3})

and

\int_{R^{3}} ϕ_{u_{n}}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The next lemma shows that the functional N and

N^{'}

possesses

B L

‐splitting property which is similar to the well‐known Brezis–Lieb lemma (). Lemma ([, Lemma 2.4]) Assume that

2 s + 2 t > 3

. Let

u_{n} ⇀ u

H^{s} ({double-struckR}^{3})

and

u_{n} \to u

a.e.in

R^{3}

.…”

Section: Preliminariesmentioning

confidence: 99%

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The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

Yang

Zhao

2019

Mathematische Nachrichten

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In this paper, we study the following critical fractional Schrödinger–Poisson system trueright115.20007pt{ε2s(−Δ)su+Vfalse(xfalse)u+ϕu=Pfalse(xfalse)ffalse(ufalse)+Qfalse(xfalse)|u|2s∗−2u,indouble-struckR3,ε2t(−Δ)tϕ=u2,indouble-struckR3,where ε>0 is a small parameter, s∈(34,1),t∈false(0,1false) and 2s+2t>3, 2s∗:=63−2s is the fractional critical exponent for 3‐dimension, Vfalse(xfalse)∈C(double-struckR3) has a positive global minimum, and Pfalse(xfalse),Qfalse(xfalse)∈C(double-struckR3) are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials V,P and Q as the concentration position of these ground state solutions as ε→0+. Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.

show abstract

“…A fractional Schrödinger–Poisson system with

V = 0

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof The proof is similar to the proof of Lemma 3.3 in . For the sake of completeness, we give the details here.…”

Section: The Autonomous Problemmentioning

confidence: 86%

“…The properties of the function

ϕ_{u}^{t}

are given in the following lemma (see [, Lemma 2.3]). Lemma If

4 s + 2 t \geq 3

, then for any

u \in H^{s} ({double-struckR}^{3})

, we have (i)

ϕ_{u}^{t} \geq 0

; (ii)

ϕ_{u}^{t} : H^{s} ({double-struckR}^{3}) \to {scriptD}^{t, 2} ({double-struckR}^{3})

is continuous and maps bounded sets into bounded sets; (iii)

\int_{R^{3}} ϕ_{u}^{t} u^{2} d x \leq {C ∥ u ∥}_{\frac{12}{3 + 2 t}}^{4} \leq C {∥ u ∥}^{4}

; (iv)If

u_{n} ⇀ u

H^{s} ({double-struckR}^{3})

, then

ϕ_{u_{n}}^{t} ⇀ ϕ_{u}^{t}

{scriptD}^{t, 2} ({double-struckR}^{3})

; (v)If

u_{n} \to u

H^{s} ({double-struckR}^{3})

, then

ϕ_{u_{n}}^{t} \to ϕ_{u}^{t}

{scriptD}^{t, 2} ({double-struckR}^{3})

and

\int_{R^{3}} ϕ_{u_{n}}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The next lemma shows that the functional N and

N^{'}

possesses

B L

‐splitting property which is similar to the well‐known Brezis–Lieb lemma (). Lemma ([, Lemma 2.4]) Assume that

2 s + 2 t > 3

. Let

u_{n} ⇀ u

H^{s} ({double-struckR}^{3})

and

u_{n} \to u

a.e.in

R^{3}

.…”

Section: Preliminariesmentioning

confidence: 99%

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The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

Yang

Zhao

2019

Mathematische Nachrichten

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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%

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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Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

Tang

Yang

2021

Mathematische Nachrichten

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In this paper, we consider the following fractional Kirchhoff equation with critical nonlinearity trueright150.0pt()a+b∫R3||(−Δ)s2u2dxfalse(−normalΔfalse)su+V(x)u=Q(x)f(u)+false|ufalse|2s∗−2u,0.16emx∈R3,where a,b>0, false(−normalΔfalse)s is the fractional Laplace operator with s∈(34,1), V vanishes at infinity and 2s∗=63−2s. Under appropriate assumptions on f, we prove the existence of positive solution by using the variational method.

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Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent

Cited by 153 publications

References 65 publications

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

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

Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

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