scholarly journals Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four

2016 ◽  
Vol 15 (5) ◽  
pp. 1841-1856 ◽  
Author(s):  
Rui-Qi Liu ◽  
Chun-Lei Tang ◽  
Jia-Feng Liao ◽  
Xing-Ping Wu
Keyword(s):  
Positive Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Critical Nonlinearities ◽  
Kirchhoff Type Problem

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.

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.

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