Weighted Hardy-sobolev Critical Exponent Elliptic Equations Of Positive Solutions For The Existence Of Multiple Solutions

In this paper,firstly,a sequence of infinitely many arbitrarily small solutions converging to zero are obtained for a class of semilinear elliptic equations with critical weighted Hardy-Sobolev exponents by using the symmetric mountain pass lemma,the variational methods and some analysis techniques. Secondly,some solutions are obtained for a class of singular semilinear elliptic equations with critical weighted Hardy-Sobolev exponents by using the mount ain pass lemma,the strong maximum principle,the variational methods,and some analysis techniques.Firstly,we consider the following semilinear elliptic problem with Dirichlet boundary value conditions whereΩis an open bounded domain in RN(N≥3)with smooth boundary (?)Ωand 0∈Ω,0≤a<(?),μ(?)(N-2)2/4,0≤μ<((?)-a)2,2Na/N-2≤s<2(1+a), f∈C(Ω×R,R),2*(a,s)(?)2(N-s)/N-2(1+a),λis a positive parameter.Then we can obtain the following main results:Theorem 1 Suppose that 0≤a<(?),0≤μ<((?)-a)2,2Na/N-2≤s 2(1+a).λ>0.Assume that f(x,u)satisfies the following conditions(f1)f(x(?)-u)=-f(x,u)for all u∈R; There then existsλ*>0 such that for anyλ∈(0,λ*),problem(P1)has a sequence of non-trivial solutions un tending to zero as n�'∞.Remark 1 Theorem in the present paper generalize the results in [1] where the author only studied the case as a=0 with general form f(x,u)in suitable conditions.Secondly,we consider the following singular semilinear elliptic problem with Dirichlet boundary value conditions whereΩis an open bounded domain in RN(N≥3)with smooth boundary aQ and 0∈Ω,0≤a<(?),μ(?)(N-2)2/4,0≤μ<((?)-a)2,2Na/N-2≤s<2(1+a). 0≤σ<2(1+a).f∈C(Ω×R,R),2*(a,s)(?)2(N-s)/N-2(1+a).F(x,t)is a primitive function of f(x,t) defined by F(x,t)=f0tf(x,s)ds for x∈Ω,t∈R.Then we can obtain the follwing main results:Theorem 2 Suppose that 0≤a<(?),0≤μ<((?)-a)2-1,2Na/N-2≤s< 2(1+a),0≤σ<2(1+a),(f5)There exists a constantρ,ρ>2,such that 0<ρF(x,t)≤f(x,t)t for all x∈Ω,t∈R+\{0}.Assume that 2*=2N/N-2,β=(?) andγ=(?)-a+β.Then problem(P2)has at least a positive solution.Corollary 1 Suppose that N≥4(1+a),0≤a<(?),0≤μ<((?)-a)2-(1+a)2,2Na/N-2≤s<2(1+a).0≤σ<2(1+a).Assume that(f1)and(f2)hold. Then problem(P2)has at least a positive solution.Theorem 3 Suppose that 0≤a<(?).0≤μ<((?)-a)2,2Na/N-2≤s 2(1+a),0≤σ<2(1+a), (f7)There exists a constantρ,ρ>2,such that 0<ρF(x,t)≤f(x,t)tfor all x∈Ω,t∈R\{0}.Assume that(2)holds.Then problem(P2)has at least two distinct nontrivial solutions.Corollary 2 Suppose that N≥4(1+a):0≤a<(?),0≤μ≤((?))2-(1+a)2,2Na/N-2≤s<2(1+a),0≤σ<2(1+a).Assume that (f6) and (f7)hold. Then problem (P2) has at least two distinct nontrivial solutions.Remark 2 Theorems in the present paper generalize the results in [2] where the author only studied the case asσ=0 with general form f(x,t).Moreover, Theorem 1 also generalizes Theorem 1.1 in [3] where the author only considered the special situation that a=0 and f(x,t)=λ|t|q-1t with suitable q.