Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents

In this paper, a class of semi-linear elliptic equations involving Hardy-Sobolev critical exponents has been investigated. This problem comes from the consideration of standing waves in the anisotropic Schr\"odinger equation, also it is very important in the field of hydrodynamics, glaciology, quantum field theory and statistical mechanics. By a detailed estimation for the extremum function and using Mountain Pass Lemma with (PS) ₂ conditions, the existence of positive solutions has been obtained. On the other hand, by establishing Pohozaev-type identity and using the properties of Bessel function, the nonexistence of positive solution also has been obtained. These results are extensions of E. Jannelli's research (1. A-1. CEJ).