Existence And Multiplicity Of Solutions For Semilinear Elliptic Equations With Hardy Terms And Sobolev-Hardy Critical Exponents

This dissertation investigates a class of semilinear elliptic equations Existence and multiplicity of solutions are studied by the variational methods and some analysis techniques. The above equations have Hardy terms and critical Sobolev-Hardy exponents. WhereΩis an open bounded domain in R~N(N≥3) with smooth boundary (?)Ωand is the Hardy-Sobolev critical exponent and 2~*=2~*(0)=2N/N-2 is the Sobolev critical exponent.Firstly, forλ=1, assume that f satisfies: for someρ＞2, a detailed analysis on the (PS) sequence of the variational functionals COlresponding to the equations is given and a local compactness result is obtained. Applying this compactness result and the Mountain Pass Lemma, this thesis proves the existence and multiplicity of solutions of the above equations.Secondly, this thesis proves the existence of two positive solutions by the achieving functions of the best Sobolev-Hardy constant, Ekeland's variational principle and the Mountain Pass Lemma under the following conditions: f is nondecreasing with respect to the second variable or nondecreasing in suitable range, andλ∈(0,λ~*), whereλ~*＞0. At last. we generalize equation to p-Laplacian and prove the existence of nontrivial solutions under suitable conditions.