Abstract:

In this article, we use variational method and the critical point theory to study the existence of multiple solutions for a class of Schrödinger-Kirchhoff equation involving the fractional $p$-Laplacian operator $M\bigg(\iint_{ {\ {\Bbb R} }^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg) (-\Delta)_p^{s} u+V(x)|u|^{p-2}u=f(x, u)+\lambda h(x)|u|^{r-2}u, \ x\in {\ {\Bbb R} }^N, $ where $ \lambda\in \ {\Bbb R} , 0<s<1<r<p<2, ps<N, (-\Delta)_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 to 0.

Key words: Schrödinger-Kirchhoff equation, Fractional p-Laplacian operator, Symmetric mountain pass theorem, Dual fountain theorem