Infinitely Many Solutions For A Class Of Fractional Elliptic Problems

Recently,the study of mechanics,process control areas,ecological and economic system,circulatory system,and epidemiology indicates that more and more scholars have paid attention to the fractional partial differential equation,since the fractional partial differential equations can describe the actual phenomena from a global perspective.On the other hand,since Frederick researched elliptic equations by using functional approach in 1934,the study of differential equations has obtained fruitful results.After several decades of developments,analysis methods such as fixed point theory,variational method and critical point theory have been an important tool in studying differential equation.In recent years,many scholars discussed fractional elliptic equations by using relatively sophisticated nonlinear analysis methods,and achieved many good results.In this paper,we consider the existence of infinitely many solutions to a quasi-linear problem based on fractional Sobolev space by using variational method and critical point theory.where(?)is a nonlocal and nonlinear operator and p ?(1,?),s ?(0,1).We study problem(P)in two situations:if f(x,u)is sublinear,then we get infinitely many solutions for(P)by using the Clark's theorem;if f(x,u)is superlinear,we obtain infinitely many solutions of the problem(P)by using the Fountain theorem.