Positive Solutions For Two Classes Of Semilinear Elliptic Equations With Perturbation, Hardy Term And Critical Weighted Hardy-Sobolev Exponents

In this paper, we study the following semilinear elliptic equations with pertur-bation, Hardy term and critical weighted Hardy-Sobolev exponents, where Ω is an open bounded domain in RN (N> 3) with smooth boundary (?)Ω is the Hardy-Sobolev critical exponent. And note that is the Sobolev critical exponent. Besides, assume that F(x,t)= ∫0sf(x,s)ds. Firstly, we consider that function f∈C(Ω×R,R) does not satisfy the (AR) condition, but it satisfies the non-quadratic and supercritical growth condition. We assume that function f satisfies the following conditions.and there exists a constant B,D uniformly for almost everywhere x∈Ω;(f3) There exist constant C and ρ>2, such that F(x,t)≥ Ctρ for any μ|x|-2(1+α)., then problem (P) has at least one positive solution.Secondly, we consider that function f∈C(Ω×R, R) with subcritical growth condition and satisfies the condition that Brezis and Nirenberg given in their paper0, uniformly for x∈Ω. there exists a positive constant a, a nonemp-ty open subset ω with 0∈ω(?)Ω and a nonempty open interval I(?)(0,+∞), so that f(x,t)≥0 for almost everywhere x∈ω and for all t≥0 and f(x,t)≥σ>0 for almost everywhere x∈ω and for all t∈I; there exist positive constants σ, D3 and a nonempty open subset ω with 0∈ω(?)Ω, so that f(x,t)≥0 for almost everywhere 0∈ω and for all t≥0; f(x,t)≥σt for almost everywhere 0∈ω and for all t∈[0, D3] or f(x,t)≥σt for almost everywhere 0∈ω and for all f∈[D3,+∞];(c) For 3(1+α)≥N<2α+2+2(?), there exists a nonempty open subset ω with 0∈ω(?)Ω, so that f(x, t)≥0 for almost everywhere 0∈ω and forThen problem (P) has at least one positive solution for every f satisfies the condition (f4) and under one of the conditions (a), (b) or (c).