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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

Soohyun Bae 1,

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.
Keywords: Semilinear elliptic equation, separation, singular solution, positive solution, stability..
Mathematics Subject Classification: Primary: 35J61, 35B09, 35B40; Secondary: 35B3.
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), 77-82.

[2]

S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbb{R}^{N}$, Discrete and Conti. Dyn. Syst., 33 (2013), 555-577.

[3]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$, J. Differential Equations, 194 (2003), 460-499.

[4]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Differential Equations, 185 (2002), 225-250.

[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), 271-297.

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.

[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), 357-379.

[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), 485-506.

[9]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of non-linear elliptic equations, Comm. Pure Appl. Math., 23 (1981), 525-598.

[10]

Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations, 13 (2008), 601-640.

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.

[12]

P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations, 241 (2007), 184-205.

[13]

N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.

[14]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p=0$ in $\mathbb{R}^{N}$, J. Differential Equations, 95 (1992),304-330.

[15]

Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924.

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.

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), 77-82.

[2]

S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbb{R}^{N}$, Discrete and Conti. Dyn. Syst., 33 (2013), 555-577.

[3]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$, J. Differential Equations, 194 (2003), 460-499.

[4]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Differential Equations, 185 (2002), 225-250.

[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), 271-297.

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.

[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), 357-379.

[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), 485-506.

[9]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of non-linear elliptic equations, Comm. Pure Appl. Math., 23 (1981), 525-598.

[10]

Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations, 13 (2008), 601-640.

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.

[12]

P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations, 241 (2007), 184-205.

[13]

N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.

[14]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p=0$ in $\mathbb{R}^{N}$, J. Differential Equations, 95 (1992),304-330.

[15]

Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924.

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.

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