Wang, L.-L.; Fan, Y.-H. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents. Mathematics2024, 12, 1616.
Wang, L.-L.; Fan, Y.-H. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents. Mathematics 2024, 12, 1616.
Wang, L.-L.; Fan, Y.-H. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents. Mathematics2024, 12, 1616.
Wang, L.-L.; Fan, Y.-H. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents. Mathematics 2024, 12, 1616.
Abstract
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\"{o}dinger 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 $\left( PS\right) _{c}$ 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 (\cite[Theorem 1.A-1.C]{EJ}).
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.