具对数非线性的分数阶p-Kirchhoff型方程基态解的存在性

The Existence of Ground-State Solutions for the Fractional p-Kirchhoff Type Equation with Logarithmic Nonlinearity

具对数非线性的分数阶p-Kirchhoff型方程的研究是Kirchhoff型方程研究中的热点问题之一。基于p=2的具对数非线性抛物型方程,提出了一类p≥2具对数非线性的分数阶p-Kirchhoff椭圆型方程。针对该类方程基态解的存在性问题,在变分法的理论基础上,利用分数阶Sobolev空间理论、格林公式和积分恒等式定义了具对数非线性的分数阶p-Kirchhoff型方程的弱解及对应的Nehari泛函和能量泛函,进而给出了Nehari流形,并结合对数的性质和Holder不等式以及能量泛函下确界d与Vitali微分收敛定理证明了具对数非线性的分数阶p-Kirchhoff型方程基态解的存在性。

The study of fractional order p-Kirchhoff type equations with logarithmic nonlinearity is one of the hot topics in the study of Kirchhoff type equations.Based on p=2 parabolic equation with logarithmic nonlinearity,a class of fractional p-Kirchhoff elliptic equations belong to p≥2 with logarithmic nonlinearity is proposed.Aiming at the existence of the ground-state solution of this type of equation,on the basis of the theory of variational method,the fractional Sobolev space theory,Green’s formula,and integral identity are used to define the weak logarithmic nonlinearity of the fractional p-Kirchhoff equation solution and the corresponding Nehari functional and energy functional,furthermore,the Nehari manifold is given,combined with the properties of the logarithm,Holder’s inequality,the lower bound d of the energy functional and the Vitali differential convergence theorem,the existence of the ground-state solution for the fractional p-Kirchhoff equation with logarithmic nonlinearity is proved.