Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form −normalΔu+ϕu=u5+λfalse|x|βuγ,in1emnormalΩ,−normalΔϕ=u2,in1emnormalΩ,u>0,in1emnormalΩ,u=ϕ=0,on1em∂normalΩ, where 0 ∈ Ω is a smooth bounded domain in R3, γ∈false(0,1false),0≤β<5+γ2, and λ > 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained.

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“…Similar to the paper, 32 we can prove that (3.3) holds true. Therefore, from (3.2) and Lemma 2.1, there holds…”

Section: Existence Of a Second Positive Solution Of System (11)supporting

confidence: 76%

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

Lei

Liao

2019

Math Methods in App Sciences

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“…In the case

M \equiv 1

and

s \in (0, 1)

similar results for fractional Laplacian has been studied in using the harmonic extension technique introduced by Caffarelli and Silvestre in . In the case

M ≢ 1

and

s = 1

, there is a lot of work addressed by many researchers, see and references therein. Recently in , the authors have shown the multiplicity result for Kirchhoff type problems, without assuming any sign changing weight, with the restriction on the dimension as well as the restriction on the coefficient of the Kirchhoff term.…”

Section: Introductionmentioning

confidence: 95%

“…In the case

M ≢ 1

and

s = 1

R^{3}

with

f (x) \equiv 1

- (0true a + ε \int_{normalΩ} {| \nabla u |}^{2} d x) Δ u false(x false) = u^{5} + λ u^{q - 1}, u > 0 in Ω, u = 0 on \partial Ω,

where

Ω \subset {double-struckR}^{3}

is smooth bounded domain,

a > 0, ε > 0

is sufficiently small and

λ > 0

is a positive parameter.…”

Section: Introductionmentioning

confidence: 99%

Fractional Kirchhoff problem with critical indefinite nonlinearity

Mishra

2018

Mathematische Nachrichten

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We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form (where Ω ⊂ ℝ is a smooth bounded domain, 0 < < 1 and 1 < < 2. Here is the Kirchhoff coefficient and 2 * = 2 ∕( − 2 ) is the fractional critical Sobolev exponent. The parameter is positive and the ( ) is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results. K E Y W O R D S critical exponent, fractional Laplacian, Kirchhoff type problem M S C ( 2 0 1 0 ) 35A15, 35B33, 35R11

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“…Such equations also appear in biological systems where the function u describes a phenomenon which depends on the average of itself (such as population density), refer [2,3] and references therein. We cite [9,13,14,22,32,20] as references where the Kirchhoff equations have been treated by variational methods, with no attempt to provide the complete list.…”

Section: Introductionmentioning

confidence: 99%

n-Kirchhoff–Choquard equations with exponential nonlinearity

Arora¹,

Giacomoni²,

Mukherjee

et al. 2019

Nonlinear Analysis

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This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem (P λ,M ) below.

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Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity

Cited by 51 publications

References 27 publications

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

Fractional Kirchhoff problem with critical indefinite nonlinearity

n-Kirchhoff–Choquard equations with exponential nonlinearity

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