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On the uniqueness of singular solutions for a Hardy-Sobolev equation
| 1. | Department of Mathematics, National Central University, Chung-Li 32001, Taiwan |
| 2. | Department of Mathematics, Tamkang University, Tamsui 25137, Taiwan, Taiwan |
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Under a Creative Commons license
$\Delta u+u^p+\frac{u^{2^*(s)-1}}{|x|^s}=0 $ in $B_1 \setminus \left \{ 0 \right \},$
where $p>1, 0 < s < 2, 2^*(s)=\frac{2(n-s)}{n-2}$, $n\geq 3$ and $B_1$ is the unit ball in $ R^n$ centered at the origin. We prove that if $p>\frac{n+2}{n-2}$ then such solution is unique.
References:
| [1] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. |
| [2] |
F. Catrina and Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequality: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-258. |
| [3] |
J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities on domains with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 (2010), 401-432. |
| [4] |
N. Ghoussoub and X.S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linaire 21 (2004), 767-793. |
| [5] |
C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 (2010), 1816-1849. |
| [6] |
C.-S. Lin and Z.Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc. 132 (2004), 1685-1691. |
show all references
References:
| [1] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. |
| [2] |
F. Catrina and Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequality: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-258. |
| [3] |
J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities on domains with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 (2010), 401-432. |
| [4] |
N. Ghoussoub and X.S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linaire 21 (2004), 767-793. |
| [5] |
C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 (2010), 1816-1849. |
| [6] |
C.-S. Lin and Z.Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc. 132 (2004), 1685-1691. |
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