Existence And Multiplicity Of Solutions For Some Elliptic Equations With Hardy Potential And Hardy-Sobolev Critical Exponents

In this paper,we study some elliptic equations with Hardy potential and hardy-Sobolev citical exponents.The existence and multiplicity of solutions are established.Our method relies upon variational principle and some analysis techniques.First of all,we research a elliptic equation with two Hardy-Soboolev critical exponents as following:where ? is an open bounded domain in RN(N>3)with C2 boundary(?)? and Sobolev critical exponent and 2*(0)= 2*= 2N/N-2 is the Sobolev critical exponent,?>0 is a real parameter.By Ekeland's variational principle,we prove the existence of the first solution.Then,combining splitting lemma with Mountain Pass Lemma,we find the existence of the second solution.Secondly,we consider a elliptic equation with weight Hardy-Sobolev critical exponent:where ? is an open bounded domain in RN(N>3)with C2 boundary(?)? and 2N/N-2(1+a-b)is the critical weighted Hardy-Sobolev exponent and 2*(?)p(a,a)=2N/N-2 is the critical Sobolev exponent,?>0 is a real parameter and f ? C(?×R+,R).The existence and multiplicity of solutions are established.Our method relies upon Ekeland's variational principle,Mountain Pass Lemma and strong maximum princi-ple.At last,we study Schrodinger as following:where N?31,0??<?(?)(N-1)2/4,0 ? s<2.V(x)is a given potential,k is a nonnegative constant,K(x)satisfies certain assumptions,f(x)(?)0 is some given function and satisfies f(x)? H-1,H-1 represents the dual space of H and we define:where Making use of Ekeland's variational principle and Nehari manifold to prove the existence of the least energy solution with k ? 0.Subsequently,the existence of the second solution is established by Mountain Pass Lemma as k = 0.