Existence Of Positive Solutions For Elliptic Problem Involving Hardy-Sobolev-Maz’ya Terms And Hardy-Sobolev Critical Exponents

We consider the following singular semilinear elliptic problem with Hardy-Sobolev-Maz’ya terms(?)where f∈C((?)×R,R), Ω is a smooth bounded domain in RN = Rk × RN-k with N ≥ 3 and 2 ≤ k < N, a point x∈RN is denoted as x = (y,z)∈Rk×RN-k and (0, z0) ∈Ω, 0≤μ<(?)=(k-2)2/4 for k > 2,μ = 0 for k = 2. The so-called Hardy-Sobolev critical exponents are denoted by 2*(s) =2(N-s)/N-2w 2* = 2*(0) =1n/n-2 are the Sobolev critical exponents.In this thesis, under the different conditions on f(x,u), we will estimate the critical value and prove the local (PS) condition, then get the existence of posi-tive solutions using the Mountain Pass Lemma and the strong maximum principle. There are our results:(1) When f(x,u) is a linear term, that is, f(x,u) = u, one has Theorem 1 Assume that N>4,(?),Then for all λ∈(0,λ1)，Problem (H) has at least one positive,where λ1 is the first eigenvalue of -△-μ|y|-2 and μ*=(?). (2) When λ = 1 and f(x,u) is asymptotically linear, thenTheorem 2 Suppose N > 2k - 2 - 2√((k-2)2-4μ)(1/1). 2 <k< N. 0 <μ< (?) and s = 2-(N-2)/N-k+√((k-2)2-4μ)(1/2).f∈C((?)×R+,R) satisfies(f1) (?) uniformly for x∈(?),where 0 <γ <λ1 and λ1 is the first eigenvalue of -(?)-μ|y|-(f2) (?) uniformly for x∈(?),(f3) there exist positive constant a, a nonempty open subset w with (0, z0)∈w(?)Ω and a nonempty open interval I C (0,+∞), so that f(x,t) > 0 for almost everywhere x G uj and for all t ≥ 0, and f(x, t) >σ> 0 for almost everywhere x G uj and for all t ∈I.Then problem (H) admits at least one positive solution.(3) When A =λ and f(x,u) is superlinear, thenTheorem 3 Suppose N ≥ 2k -2-2((k-2)2-4μ)(1/2), 2 ≤k< N, 0 ≤μ <(?) and s = 2-(N-2)/N-k+((k-2)2-4μ)(1/2)=.f∈G((?) ×R+,R) satisfies (f3) and(f4) (?)uniformly for x∈(?)(f5) (?)uniformly for x∈(?)(f6)(?)uniformly for x∈(?)(f7) |f(x,t)|τ ≤ a1F(x,t)|t|τ for some a1 > 0, τ > 1 and (x,t) ∈ (?)×R+ with t large enough, where F(x,t) =1/2f(x,t)t -F(x,t).Then problem (H) possesses at least a positive solution.Theorem 4 Suppose N ≥3, 2≤k < N and 0 ≤μ<(?). f ∈C((?) ×R+, R) satisfies (f2),(f4) and(f8) There exists a nonempty open subset Ω0(?)Ω with (0,z0) ∈Ω0, such that f(x,t) ≥ 0 for almost everywhereb x ∈Ω and all t > 0, and f(x,t) > 0 for almost x∈Ω0 and all t > 0.Then there exists (?)* > 0, such that problem (H) admits at least one positive solution for all λ ≥(?)*.