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- Title
Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents.
- Authors
Wang, Lin-Lin; Fan, Yong-Hong
- Abstract
The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents − Δ u − μ u x 2 = u 2 * s − 2 x s u + g (x , u) , x ∈ Ω ∖ 0 , u = 0 , x ∈ ∂ Ω have been investigated, where Ω is an open-bounded domain in R N N ≥ 3 , with a smooth boundary ∂ Ω , 0 ∈ Ω , 0 ≤ μ < μ ¯ : = N − 2 2 2 , 0 ≤ s < 2 , and 2 * s = 2 N − s / N − 2 is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on g , by a detailed estimation of the extremum function and using mountain pass lemma with P S c conditions, we obtained that: (a) If μ ≤ μ ¯ − 1 , and λ < λ 1 μ , then the above problem has at least a positive solution in H 0 1 Ω ; (b) If μ ¯ − 1 < μ < μ ¯ , then when λ * μ < λ < λ 1 μ , the above problem has at least a positive solution in H 0 1 Ω ; (c) if μ ¯ − 1 < μ < μ ¯ and Ω = B (0 , R) , then the above problem has no positive solution for λ ≤ λ * μ. These results are extensions of E. Jannelli's research ( g (x , u) = λ u ).
- Subjects
CRITICAL exponents; QUANTUM field theory; STATISTICAL mechanics; SCHRODINGER equation; STANDING waves
- Publication
Mathematics (2227-7390), 2024, Vol 12, Issue 11, p1616
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math12111616