On Hardy Equations

This dissertation investigates singular semilinear elliptic equations, which have singular coefficients and critical Sobolev-Hardy exponents or critical Sobolev exponents. Firstly, this paper deal with this problemwhere 2^* = 2N/(N - 2) is the critical Sobolev exponent, . Through a compactness analysis of the functional corresponding to the problems (a), we obtain the existence of positive solutions for this problem under certain assumptions on a(x) and k(x): andwhere 2^* = 2N/(N - 2) is the limiting exponent for the embedding of H¹(Ω) into L^p(Ω), andλ∈R¹ are parameters,α(x)∈C((?)Ω) ,α(x)≥0. Through a compactness analysis of the functional corresponding to the problem (6), we obtain the existence of positive solutions for this problem under different assumptions on the parametersμ,λand the fact that 0∈Ωor 0∈(?)Ω. The main difference of problem (a) and (b) is in that the domain ( bounded or unbounded ) and the boundary condition (Dirichlet or Rabin). And we must take different method to deal the two problem, also this difference of the two problem induces very different blow up phenomena and results.Secondly, for the following problem related to the Caffarelli-Kohn-Nirenberg inequalitieswhere a = b < 0.λandηare real constants. We obtain some existence or nonexistence results for this problem. For nonhomogeneous Neumann boundary value problem of the typewhereΩis a bounded domain in R^N with smooth boundary,is the p-Laplacian operator,and A is a real constant, we have theexistence, nonexistence and multiplicity of positive solutions.Finally we are concerned the following problemwhere N > 2, K(x) is a bounded, continuous function satisfying some conditions. D_G^1,2 (R^N) is an appropriate Sobolev space of G-symmetric functions. is the critical Sobolev-Hardy exponent, and 0≤s<2, 0<μ<μ|-= ((N₂)/2)² .We obtain the existence of G—symmetry solution of this problem.