Version 1
: Received: 26 March 2024 / Approved: 27 March 2024 / Online: 27 March 2024 (15:07:38 CET)
How to cite:
Wang, L.; Fan, Y. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents. Preprints2024, 2024031680. https://doi.org/10.20944/preprints202403.1680.v1
Wang, L.; Fan, Y. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents. Preprints 2024, 2024031680. https://doi.org/10.20944/preprints202403.1680.v1
Wang, L.; Fan, Y. Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents. Preprints2024, 2024031680. https://doi.org/10.20944/preprints202403.1680.v1
APA Style
Wang, L., & Fan, Y. (2024). Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents. Preprints. https://doi.org/10.20944/preprints202403.1680.v1
Chicago/Turabian Style
Wang, L. and Yong-Hong Fan. 2024 "Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents" Preprints. https://doi.org/10.20944/preprints202403.1680.v1
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.
