中文摘要：

该文研究带凹凸项的分数阶Laplace方程{（-Δ）^su=λa（x）｜u｜^q-2u＋b（x）｜u｜^p-2u在Ω上,u=0 在Rn／Ω上解的存在性,其中Ω是Rn中的有界区域,s∈（0,1）,q∈（1,2）,p∈（2,2s^＊],2s^＊=2n/n-2s,n＞2s,λ＞0,a（x）和b（x）都是有界连续函数,且b（x）非负、a（x）变号.应用山路引理,证明了方程在临界和次临界情形下,至少有一个非负非平凡解;而且,利用喷泉定理,证明了方程在次临界情形下有无穷多个解.

英文摘要：

In this paper, we investigate the existence of solutions for the following equation with the fractional Laplacian （-△）8 and concave-convex nonlinearities,{（-Δ）su=λa（x）｜u｜q-2u＋b（x）｜u｜p-2u inΩ inR^n/Ω where,s∈（0,1）,q∈（1,2）,p∈（2,2＊s],2＊s=2n/n-2s,n〉2s,λ〉0,Ωis a bounded domain of R^n, a（x） and b（x） are bounded continuous with b（x） ≥ 0 and a（x） changes signs. We not only prove the existence of nontrivial nonnegative solutions by the Mountain Pass LemIna when p is subcritical and critical, but also, by using Fountain Theorem, we obtain infinitely many solutions for the subcritical case.