# American Institute of Mathematical Sciences

• Previous Article
Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains
• PROC Home
• This Issue
• Next Article
Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem
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), 77-82. Google Scholar [2] S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbbR^n$, Discrete and Conti. Dyn. Syst., 33 (2013), 555-577.  Google Scholar [3] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$, J. Differential Equations, 194 (2003), 460-499.  Google Scholar [4] S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$, J. Differential Equations, 185 (2002), 225-250. 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), 271-297.  Google Scholar [6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  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), 357-379.  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), 485-506.  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), 525-598.  Google Scholar [10] Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations, 13 (2008), 601-640.  Google Scholar [11] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  Google Scholar [12] P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations, 241 (2007), 184-205.  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), 233-272.  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),304-330.  Google Scholar [15] Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbbR^n$, Duke Math. J., 57 (1988), 895-924.  Google Scholar [16] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.  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), 77-82. Google Scholar [2] S. Bae, On the elliptic equation $\Deltau+Ku^p=0$ in $\mathbbR^n$, Discrete and Conti. Dyn. Syst., 33 (2013), 555-577.  Google Scholar [3] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$, J. Differential Equations, 194 (2003), 460-499.  Google Scholar [4] S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$, J. Differential Equations, 185 (2002), 225-250. 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), 271-297.  Google Scholar [6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  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), 357-379.  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), 485-506.  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), 525-598.  Google Scholar [10] Q. Jin, Y. Li and H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations, 13 (2008), 601-640.  Google Scholar [11] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  Google Scholar [12] P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations, 241 (2007), 184-205.  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), 233-272.  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),304-330.  Google Scholar [15] Y. Li and W.-M. Ni, On conformal scalar curvature equation in $\mathbbR^n$, Duke Math. J., 57 (1988), 895-924.  Google Scholar [16] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.  Google Scholar
 [1] Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335 [2] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193 [3] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [4] Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053 [5] Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166 [6] Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110 [7] Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227 [8] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [9] Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313 [10] José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653 [11] Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 [12] Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 [13] Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085 [14] Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3331-3355. doi: 10.3934/cpaa.2021108 [15] Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122 [16] Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399 [17] Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198 [18] Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123 [19] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [20] Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

Impact Factor: