Existence Of Multiple Positive Solutions For A Singular Nonlinear Elliptic Equation Involving Hardy-Sobolev-Maz’ya Term

In this paper,we consider the following singular nonlinear elliptic equation involving Hardy-Sobolev-Maz’ya terms where RN=Rk×Rn-k,2≤k<N,0≤λ<(k-2)2/4when k>2,λ=0when k=2,0≤t<2,pt=N+2-2T/N-2and μ>0is a parameter,f(x)≥0is smooth and f(x)(?)0.A point x∈RN is denoted as x=(y,z)∈Rk×RN-k and(0,z0)∈Ω(?)RN is a bounded dom ain with smooth boundary.In this article we mainly want to show the problem has at least two positive solutions when μ is small enough and pt=1+In order to prove the result,we first transform the problem to an equivalent equation.Then we consider the new problem and prove that the problem possesses at least two solutions by using the sub-solutions and super-solutions argument and the varriant of Mountain pass theorem without the(PS)condition respectively.