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2013, 2013(special): 123-128. doi: 10.3934/proc.2013.2013.123

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

Received  September 2012 Revised  February 2013 Published  November 2013

In this paper, we consider the positive singular solutions for the following Hardy-Sobolev equation
                        $\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.
Citation: Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
References:
[1]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math. 36 (1983), 36 (1983), 437.

[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), 54 (2001), 229.

[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), 197 (2010), 401.

[4]

N. Ghoussoub and X.S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincaré Anal. Non Linaire 21 (2004), 21 (2004), 767.

[5]

C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg,, J. Funct. Anal. 259 (2010), 259 (2010), 1816.

[6]

C.-S. Lin and Z.Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities,, Proc. Amer. Math. Soc. 132 (2004), 132 (2004), 1685.

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), 36 (1983), 437.

[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), 54 (2001), 229.

[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), 197 (2010), 401.

[4]

N. Ghoussoub and X.S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincaré Anal. Non Linaire 21 (2004), 21 (2004), 767.

[5]

C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg,, J. Funct. Anal. 259 (2010), 259 (2010), 1816.

[6]

C.-S. Lin and Z.Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities,, Proc. Amer. Math. Soc. 132 (2004), 132 (2004), 1685.

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