一类分数阶Schrödinger-Kirchhoff方程多重解的存在性  

Existence of Multiple Solutions for a Class of Fractional Schrödinger-Kirchhoff Equation

作　　者：李建利 李安然[2] 魏重庆 李刚[3] Li Jianli;Li Anran;Wei Chongqing;Li Gang(Department of Applied Mathematics,Taiyuan Institute,Taiyuan 030032;School of Mathematical Sciences,Shanxi University,Taiyuan 030006;School of Mathematical Sciences,Yangzhou University,Jiangsu Yangzhou 225002)

机构地区：[1]太原学院应用数学系,太原030032 [2]山西大学数学科学学院,太原030006 [3]扬州大学数学科学学院,江苏扬州225002

出　　处：《数学物理学报（A辑）》2020年第6期1612-1621,共10页Acta Mathematica Scientia

基　　金：国家自然科学基金(11701346,11871064)。

摘　　要：利用变分方法和临界点理论讨论了一类带有分数阶p-拉普拉斯算子的Schrödinger-rKirchhoff方程多重解的存在性M(∫∫R^2N|u(x)-u(y)|^p/|x-y|^N+psdxdy)(-Δ)p^s u+V(x)|u|^p-2u=f(x,u)+λh(x)|u|^r-2u,x∈R^N,其中λ∈R,0<s<1<r<p<2,ps<N,(-Δ)p^s;表示分数阶p-拉普拉斯算子.首先,利用对称山路定理得到该方程无穷多高能量解的存在性.其次,利用对偶喷泉定理证明了上述方程有一列趋于0的负能量解.In this article,we use variational method and the critical point theory to study the existence of multiple solutions for a class of Schrodinger-Kirchhoff equation involving the fractional p-Laplacian operator M(∫∫R^2 N|u(x)-u(y)|^p/|x-y|^N+psdxdy)(-Δ)p^s u+V(x)|u|^p-2u=f(x,u)+λh(x)|u|^r-2u,x∈R^N,whereλ∈R,0<s<1<r<p<2,ps<N,(-Δ)p^s is the fractional p-Laplacian operator.Under certain assumptions,we first show the existence of multiple high energy solutions by means of symmetric mountain pass theorem.Secondly,by using dual fountain theorem,we prove that the above equation has a sequence of negative energy solution,whose energy converges to0.

关 键 词：Schrödinger-Kirchhoff方程 分数阶p-拉普拉斯算子 对称山路定理 对偶喷泉定理 

分 类 号：O176.3[理学—数学]