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2013, 2013(special): 31-39. doi: 10.3934/proc.2013.2013.31

Classification of positive solutions of semilinear elliptic equations with Hardy term

1. 

Hanbat National University, Daejeon 305-719

Received  September 2012 Revised  July 2013 Published  November 2013

We study the elliptic equation $\Delta u+\mu/|x|^2+K(|x|)u^p=0$ in $\mathbb{R}^n \setminus \left \{ 0 \right \}$, where $n\geq1$ and $p>1$. In particular, when $K(|x|)=|x|^l$, a classification of radially symmetric solutions is presented in terms of $\mu$ and $l$. Moreover, we explain the separation structure for the equation, and study the stability of positive radial solutions as steady states.
Citation: Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31
References:
[1]

S. Bae, On positive solutions of nonlinear elliptic equations with Hardy term, Mathematical analysis and functional equations from new points of view (Kyoto, 2010)., RIMS Kôkyûroku No. 1750 (2011), 1750 (2011), 77.   Google Scholar

[2]

S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbbR^n$,, Discrete and Conti. Dyn. Syst., 33 (2013), 555.   Google Scholar

[3]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 460.   Google Scholar

[4]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$,, J. Differential Equations, 185 (2002), 225.   Google Scholar

[5]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.   Google Scholar

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.   Google Scholar

[7]

D. Damanik, D. Hundertmark and B. Simon, Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators,, J. Funct. Anal., 205 (2003), 357.   Google Scholar

[8]

W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u+K u^{(n+2)/(n-2)} = 0$ and related topics,, Duke Math. J., 52 (1985), 485.   Google Scholar

[9]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of non-linear elliptic equations,, Comm. Pure Appl. Math., 23 (1981), 525.   Google Scholar

[10]

Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres,, Adv. Differential Equations, 13 (2008), 601.   Google Scholar

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.   Google Scholar

[12]

P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations,, J. Differential Equations, 241 (2007), 184.   Google Scholar

[13]

N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities,, Invent. Math., 135 (1999), 233.   Google Scholar

[14]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p=0$ in $\mathbbR^n$,, J. Differential Equations, 95 (1992), 304.   Google Scholar

[15]

Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbbR^n$,, Duke Math. J., 57 (1988), 895.   Google Scholar

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381.   Google Scholar

show all references

References:
[1]

S. Bae, On positive solutions of nonlinear elliptic equations with Hardy term, Mathematical analysis and functional equations from new points of view (Kyoto, 2010)., RIMS Kôkyûroku No. 1750 (2011), 1750 (2011), 77.   Google Scholar

[2]

S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbbR^n$,, Discrete and Conti. Dyn. Syst., 33 (2013), 555.   Google Scholar

[3]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 460.   Google Scholar

[4]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$,, J. Differential Equations, 185 (2002), 225.   Google Scholar

[5]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.   Google Scholar

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.   Google Scholar

[7]

D. Damanik, D. Hundertmark and B. Simon, Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators,, J. Funct. Anal., 205 (2003), 357.   Google Scholar

[8]

W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u+K u^{(n+2)/(n-2)} = 0$ and related topics,, Duke Math. J., 52 (1985), 485.   Google Scholar

[9]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of non-linear elliptic equations,, Comm. Pure Appl. Math., 23 (1981), 525.   Google Scholar

[10]

Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres,, Adv. Differential Equations, 13 (2008), 601.   Google Scholar

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.   Google Scholar

[12]

P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations,, J. Differential Equations, 241 (2007), 184.   Google Scholar

[13]

N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities,, Invent. Math., 135 (1999), 233.   Google Scholar

[14]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p=0$ in $\mathbbR^n$,, J. Differential Equations, 95 (1992), 304.   Google Scholar

[15]

Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbbR^n$,, Duke Math. J., 57 (1988), 895.   Google Scholar

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381.   Google Scholar

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