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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

Jann-Long Chern 1, , Yong-Li Tang 1, , Chuan-Jen Chyan 2, and  Yi-Jung Chen 2,

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.
Keywords: Hardy-Sobolev equation, singular solution, uniqueness of solutions..
Mathematics Subject Classification: Primary: 35J60; Secondary: 35A0.
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), 437-477.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 (2010), 1816-1849.  Google Scholar

[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.  Google Scholar

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