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

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.