scholarly journals Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Dechen Wang ◽  
Baoqiang Yan
Keyword(s):  
Positive Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Multiplicity Of Positive Solutions

scholarly journals Multiplicity of positive solutions for critical fractional Kirchhoff type problem with concave-convex nonlinearity

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3791-3798
Author(s):  
Chang-Mu Chu ◽  
Zhi-Peng Cai ◽  
Hong-Min Suo
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Multiplicity Of Positive Solutions

This paper is devoted to study a class of Kirchhoff type problem with critical fractional exponent and concave nonlinearity. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.

scholarly journals Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

2020 ◽  
pp. 1-38 ◽  
Author(s):  
Qilin Xie ◽  
Xu Zhang
Keyword(s):  
Variational Method ◽  
Critical Frequency ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Classical State ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

Abstract In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

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