Multiple Solutions For A Class Of Fractional P-Laplacian Problems On Bounded Domains

Recently,a great attention has been focused on the study of fractional and non-local differential operators both for the pure mathematical research and in the view of concrete application.Among them,the fractional p-Laplacian operator is a kind of non-local pseudodifferential operators.The equations involving the fractional p-Laplacian operator are used to describe the diffusion phenomenon,and it has been widely used in fluid mechanics,materials memory,biology,plasma physics,finance,chemistry and so on.In this work,we consider that multiple solutions for the fractional p-Laplacian equation as followwhere Ω is a smooth bounded domain of RN(N≥1),0<s<1<p<∞.It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti-Rabinowitz type condition on f(x,u).An alternative condition has been used in this paper,as f(x,u)also satisfied the subcritical growth condition,we use variational method and critical point theory,the existence of multiple solutions of the fractional p-Laplacian equation to be obtain.In particularly,considered about the fact that the fractional Laplacian equation has the widely practical signficance,we also turn out the existence of mulitiple solutions of the fractional Laplacian equation.