Existence And Multiplicity Of Positive Solutions For Some Singular Elliptic Equations

In this paper,by the variational principle and some analysis techniques,we study some singular elliptic equations.The existence and multiplicity of positive solutions are established.Firstly,we research the following elliptic equation with Dirichlet boundary condition:where ? is an open bounded domain in RN(N?3)with C2 boundary(?)? and(?),0?s<2,2*(s)=2(N s)/N-2 are the Hardy-Sobolev critical exponent and 2*(0)=2*=2N/N-2 is the Sobolev critical exponent,g ?C(? x R,R).By mountain pass lemma and strong maximum principle,we get the existence of the positive solution.Secondly,we consider a elliptic equation with weight Hardy-Sobolev critical exponent and inhomogeneous Neumann boundary condition:where ? is an open bounded domain in RN(N?3)with C2 boundary(?)? and#12(?)is the critical weighted Hardy-Sobolev exponent and p(a,a)=2N/N-2 is the critical Sobolev exponent,?>0 is a real parameter and g?c(?×R+,R),v denotes the unit outward normal vector to boundary(?)?,??L2((?)?)is a nonnegative function and 0(?)0.Using Ekeland,s variational principle,we get the first positive solution.Then,by Mountain Pass Lemma,the existence of the second positive solution are obtained.