The Existence Study With Hardy Singular Solutions For Semilinear Elliptic Equations

In this paper,we study the existence of solutions for three classes of semilin-ear elliptic equations with Hardy singular terms(respectively,semilinear elliptic equations involving general subcritical growth,a resonant semilinear elliptic equa-tion,semililinear elliptic problems involving critical Hardy-Sobolev exponent)via variational methods and analysis techniques.Its main contents are as follows:Firstly, in Chapter2,we consider the following Dirichlet boundary value problem where Ω is an open bounded domain in RN(N≥3) with smooth boundary aΩ, μ<μΔ(N-2)2/4,f(x,t) is continuous on Ω×R.We consider a class of elliptic partial differential equations with more general growth condition.We assume the conditions below:(F1) where2*=2N/N-2is the Sobolev critical exponent.We obtain the following theorem.Theorem1Let(F1)hold,and the following conditions(F2)一(F4)hold,(F2)There exist constants α≥1and c>0such that where G(x,t):=tf(x,t)一2F(x,t).(F3)(F4)Then equation(0.1)has a nontrivial solution for all λ>0. Next, we consider the special case λ=1of equation (0.1), that is We obtain the following theorem.Theorem2Let (F1) hold, and the following conditions hold,(F5) There exist constants θ∈(0,1/2), M>0such that(F6)(F7) where ε>0, λ1is the first eigenvalue of the operator-△-μ/|x|2with Dirichlet boundary conditions.Then equation (0.2) has a weak nontrivial solution.In Chapter3, we consider resonant problem of equation (0.2). We obtain the following theorem.Theorem3There exist constants M0>0such that where a and b are continuous functions. Suppose the following double resonant condition hold, where λκ(α) is the κth eigenvalue of the operator-△-μ/|x|2-αwith Dirichlet boundary conditions. Moreover, the following conditions are satisfied: Then equation (0.2) has a solution.In Chapter4, we consider the following semilinear elliptic problem with Hardy-Sobolev exponent:In this Chapter, we assume that k satisfies one of the following conditions:(K) κ∈L∞(RN)∩C1(RN), κ(χ)=κ{｜χ｜)=κ(r), r=|χ|, and for some α<N-s.(K’)κ∈C2(RN), κ(χ)=κ(｜χ｜)=κ(r), r=|χ|, κ(r) is T-periodic andThe main results are the following theorems.Theorem4Let (K) hold, and assume that κ(0)=0and κ(?)0. Then for|ε|small, problem (0.3) has a positive radial solution uε.Theorem5Let (K) hold, and assume that κ∈C2(RN) and κ(0)κ"(0)>0. Then for |ε|small, problem (0.3) has a positive radial solution uε.Theorem6Assume that (K) holds, and suppose that and Then for|ε|small, problem (0.3) has a positive radial solution uε.Theorem7Suppose that assumption (K’) holds, and satisfies the condition κ(0)κ"(0)>0. Then problem (0.3) has a positive radial solution uε, provided|ε|《1.Theorem8Let κ satisfy (K’), and suppose that κ(0)=0and κ(?)0. Then problem (0.3) has a positive radial solution uε,provided ｜ε｜《1.