Existence Of Solutions For Fractional Schr?dinger-poisson System

With the rapid development of the applied subjects,nonlinear partial differential equa-tions have widely been studied.Some nonlinear partial differential equations are technolog-ical tools which can be applied to solve many problems in the field of natural science and engineering.They can also be used to describe the mathematical models in daily life.In the present paper,existence of solutions for fractional Schr?dinger-Poisson system is studied.Due to the physical meaning of the system in the daily life,it is valuable to study them.In the first chapter,we studied the the existence of sign-changing solutions for fractional Schr?dinger-Poisson:where N ? 3,? ?(0,N)and p?[2,(N+?)/(N-2))V and f? C1(R,R)satisfy the following conditions:(V)V is a positive constant or V ? C(IRN,R+)such that lim|x|??(x)= ?,where R+ =[0,?).(f1)lim|s|??f'(s)/|S|2*-2 = 0,where 2*= 2N/(N-2);(f2)the function f(s)/(|s|2(p-1)s)is nonincreasing on(-?,0)and nondecreasing on(0,?)respectively,lims?f(s)/(|s|2(p-1)s)= 0 and lim|s|??f(s)/(|s|2(p-1)s)= ?.Applying quantitative lemma and topological degree theory,we get main theorem as follows:Theorem 1.3.1 Suppose the assumptions(V),(f1)and(f2)hold.Then the system(1.1.1)has at least one sign-changing solution.In the second chapter,we studied the fractional Schr?dinger-Poisson system:where a E(0,3),p ?[2,3 + ?).V and f which are periodic and continuous satisfy the following conditions:(V)V is 1-periodic with respect to xi,where i = 1,2,3,V0 = infx?R3 V(x)>0;(f1)f is 1-periodic with respect to xi,there exist q ?(4,6)and C>0 such that limt?? f(x,t)/|t|q-1 = 0,where i = 1,2,3;(f2)limt?0 f(x,t)/|t|p = 0 uniformly x? R3;(f3)lim|t|??F(x,t)|t|2p = ? uniformly x? R3,where F(x,xt)=fot f(x,s)ds;(f4)the functionf(x,t)/(|t|2(p-1)t)is nonincreasing on(-?,0)and nondecreasing on(0,?)respectively.Theorem 2.3.1 Suppose the assumptions(V),(f1)-(f4)hold.Then the system(2.1.1)has at least one ground state solution.