With Positive Solutions Of Elliptic Equations In Weighted Hardy-sobolev Critical Exponents

This dissertation investigates a class of semilinear elliptic equations with critical weighted Hardy-Sobolev exponents. Existence and multiplicity of positive solutions are studied by the variational methods and some analysis techniques.Firstly, we considered the following scmilinear elliptic equation: Here, Q is a smooth bounded domain in RN(N>3),0∈Ω, 0≤a<(?), with a≤b< a+1,λand v are positive parameters, is the Hardy-Sobolev critical exponent. Note that is the Sobolev critical exponent. f(x) is a positive measurable function, g∈C(Ω×R,R), G(x. t) is the primitive function of g(x, t) defined by for x∈Ω, t∈R. For this equation we get the first positive local minimum for the associated functional by Ekeland's variational principle. And by choosing special mountain pass and energy estimates to find that the functional satisfies the (PS)c condition on a given range, by using the Mountain Pass Theorem, we find the second positive solution. So the existence of two positive solutions are obtained.And then, we study the following semilinear elliptic equation under the dif-ferent conditions: existence and multiplicity of positive solutions are obtained by using the Mountain-Pass Lemma.