On Semilinear Singular Elliptic Equations

This dissertation investigates singular semilinear elliptic equations, which have sigular coefficients and critical Sobolev-Hardy exponents or critical Sobolev exponents. Firstly, this paper deals with the equation where a smooth bounded domain in detailed analysis on the (PS) sequence of the variational founctionals corresponding to the equations is given and a local compactness result is obtained. Applying this compactness result, this thesis proves the existence of positive solutions(Mountain-pass solutions) and the sign-changing solutions. The main difficulties are to find the functions which achieve the best constant of critical Sobolev-Hardy imbedding and to overcome the singularity of the positive solutions. By the achieving functions of the best Sobolev-Hardy constant and the linking theorem, this paper deals with the elliptic equation with critical Sobolev-Hardy exponents proves theexistence of nontrival solutions for a large range of (the first eigenvalue of the operator they are sign-changing solutions.For the nonhomogeneous equations with critical Sobolev-Hardy exponents this paper proves the existence of multiple solutions under some conditions by the achieving functions of the best Sobolev-Hardy constant, Ekeland's variational principle and the mountain-pass theorem. The singularity of the solutions needs also to be overcomed in the discussion.In the last, by the above results, this thesis studies the elliptic equations with critical Sobolev exponents proves the existence of infinitely many solutions under some conditions.