Boundary Value Problem Of Semilinear Elliptic Equations

This dissertation investigates a class of semilinear singular elliptic equations. Existence and multiplicity of solutions are studied by the variations! methods and some analysis techniques. Firstly, we considered the following elliptic equation with Dirichlet boundary condition:This equations have Hardy terms and critical Sobolev-Hardy exponents. WhereΩis an open bounded domain in R(N≥3) with smooth boundary (?)Ω., is the Hardy-Sobolev critical exponent and is the Sobolev critical exponent,λ> 0 is a real constant. A de-tailed analysis on the (PS) sequence of the variational functionals corresponding to the equations is given and a local compactness result is obtained. Applying this compactness result and the Mountain Pass Lemma, the existence result of solutions is obtained. Where, we main considered the case that 0∈(?)Ω. In this case, in order to get the solutions of the Mountain Pass typical, we need consider the properties of the curvature of (?)Ωat 0 and estimate the attain function of the Sobolev-Hardy critical case. Moreover, using the Fountain Theorem and the dual Fountain Theorem, we get the existence of infinitely numerous solutions for the above equations.Secondly, this thesis studied the following singular Neumann problems:Where, 0∈(?)Ω. Firstly, we get some new estimate to the functions which achieve the best constant of critical Sobolev-Hardy imbedding. Unlike the Dirichlet boundary problem or the case that 0∈(?)Ω, there are many difficulties to estimate the attain function of the Sobolev-Hardy critical case. Secondly, By Ekeland's variational principle and the Mountain Pass Lemma, the existence and multiplic- ity results of the positive solutions are obtained.Key words: Semilinear elliptic equation, Sobolev-Hardy critical exponents, boundary singularities, Dirichlet boundary condition, Neumann boundary condi-tion.