Existence And Multiplicity Of Solutions For Fractional Elliptic Equation With Hardy Potential And Critical Hardy-Sobolev Exponents In R~N

Fractional Laplace operator is a nonlocal operator,which has been widely used in many fields,such as finance,physics,medicine,chemistry and so on.And singular potential has physical meaning,for example quantum mechanics,molecular physics,nuclear physics,quantum cosmology,molecular physics,etc.Compared with the Partial Differential Equations with regular potential,the properties of the solutions of the equationns with sigular potential would be changed.And the singular potential has many kinds of expression,such as Coulomb potential and Hardy potential.In this paper,we muainly study the fractional elliptic equation with Hardy term(singular potential)and critical Hardy-Sobolev exponents in RN,where,0<?<2,0<s<?<N,the subcritical exponent r satisfies 1<r<2<2?*(s):=2(N-s)/N-?,2?*(s):=2(N-s)/N-? is critical Hardy-Sobolev exponents.0??<?H(?)=2??2(N+?/4)/?2(N-?/4),?H(?)is the best fractional Hardy constant on RN.k?0,V is the given potential functions.In this paper,by analyzing the equation(1),when K(x),V(x)and r satisfy some conditions,we use the Ekeland variational principle,Nehari manifold and Mountain Pass Theorem to obtain the existence and multiplicity of ground state solutions of the equation(1).The main structure of this paper is as follows:In the first chapter,we introduce background of the research goal,give the relevant preparatory knowledge and the definition and explanation of the corresponding symbols,and list the two main conclusions of this paper.In the second chapter,we first give seven lemmas.From the lemma,we can obtain that the equation(1)has a minimizing sequence.The Compactness Lemma is used to proved that the minimizing sequence has a strongly convergent subsequence whose limit is a ground state solution of the equation(1).In the third chapter,we mainly research the existence of the second solution of the equation(1)when k=0.Because when k=0,the first solution of the equation(1)is still the ground state solution and the local minimal solution.First,by giving two lemmas,lemma 3.1 and lemma 3.2,we show that the corresponding energy functional I of the equation(1)satisfies the(PS)condition.Then,we get that there is a large enough t such that I(u0+tv0)-I(u0)<0.Finally,we use Mountain Pass Theorem to prove the existence of the second solution of the equation(1).In the fourth chapter,we make a summary of the research results of this artical,and put forwarding some question which we can be studied in the future.For instance,we can consider the multiplicity of solutions to the equation(1)when k>0.During the parameter ??0,we also think about asymptotic properties of the equation which is At the same time,we can discuss the homogenization and concentration of solution of the equation(2).