Positive solutions for the Kirchhoff-type problem involving general critical growth – Part I: Existence theorem involving general critical growth

2018 ◽  
Vol 460 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Huixing Zhang ◽  
Cong Gu ◽  
Chun-Ming Yang ◽  
Jean Yeh ◽  
Juan Jiang
Keyword(s):  
Existence Theorem ◽  
Positive Solutions ◽  
Critical Growth ◽  
Type Problem ◽  
Kirchhoff Type ◽  
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.

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