# American Institute of Mathematical Sciences

February  2013, 33(2): 555-577. doi: 10.3934/dcds.2013.33.555

## On the elliptic equation Δu+K up = 0 in $\mathbb{R}$n

 1 Faculty of Liberal Arts and Sciences, Hanbat National University, Daejeon, 305-719

Received  July 2011 Revised  April 2012 Published  September 2012

This paper deals with the elliptic equation Δu+K(|x|) up = 0 in $\mathbb{R}$n\{0} when $r^{-l}K(r)$ for $l>-2$ behaves monotonically near $r=0$ or $\infty$ with $r=|x|$. By the method of phase plane, we present a new proof for the structure of positive radial solutions, and analyze the asymptotic behavior at $\infty$. We also employ the approach to classify singular solutions in terms of the asymptotic behavior at $0$. In particular, when $p=\frac{n+2+2l}{n-2}$, we establish the uniqueness of solutions with asymptotic self-similarity at $0$ and at $\infty$, and the existence of multiple solutions of Delaunay-Fowler type at $0$ and $\infty$.
Citation: Soohyun Bae. On the elliptic equation Δu+K up = 0 in $\mathbb{R}$n. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 555-577. doi: 10.3934/dcds.2013.33.555
##### References:
 [1] S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient, J. Differential Equations, 237 (2007), 159-197. doi: 10.1016/j.jde.2007.03.003. [2] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$, J. Differential Equations, 194 (2003), 460-499. doi: 10.1016/S0022-0396(03)00172-4. [3] S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Differential Equations, 185 (2002), 225-250. doi: 10.1006/jdeq.2001.4162. [4] 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. doi: 10.1002/cpa.3160420304. [5] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] W. Y. Ding and W. M. Ni, On the elliptic equation $\Delta u+Ku^(n+2)/(n-2)=0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. [7] R. H. Fowler, Further studies of Emden's and similar differential equations, Quarterly J. Math, 2 (1931), 259-288. [8] 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. doi: 10.1002/cpa.3160340406. [9] R. A. Johnson, X. Pan and Y. Yi, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302. doi: 10.1016/0362-546X(93)90132-C. [10] R. A. Johnson, X. Pan and Y. Yi, Positive solutions of super-critical elliptic equations and asymptotics, Comm. Partial Differential Equations, 18 (1993), 977-1019. doi: 10.1080/03605309308820958. [11] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269. doi: 10.1007/BF00250508. [12] 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. doi: 10.1007/s002220050285. [13] 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. doi: 10.1016/0022-0396(92)90034-K. [14] Y. Li and W. M. Ni, On conformal scalar curvature equation in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7. [15] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. [16] R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke Math. J., 99 (1999), 353-418. doi: 10.1215/S0012-7094-99-09913-1. [17] W. M. Ni, On the elliptic equation $\Delta u +K(x)u^(n+2)/(n-2)=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. [18] W. M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math., 36 (1986), 379-399. doi: 10.1002/cpa.3160390306. [19] W. M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. doi: 10.1007/BF03167899. [20] E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|)u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 124 (1993), 239-259. doi: 10.1007/BF00953068.

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##### References:
 [1] S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient, J. Differential Equations, 237 (2007), 159-197. doi: 10.1016/j.jde.2007.03.003. [2] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$, J. Differential Equations, 194 (2003), 460-499. doi: 10.1016/S0022-0396(03)00172-4. [3] S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Differential Equations, 185 (2002), 225-250. doi: 10.1006/jdeq.2001.4162. [4] 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. doi: 10.1002/cpa.3160420304. [5] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] W. Y. Ding and W. M. Ni, On the elliptic equation $\Delta u+Ku^(n+2)/(n-2)=0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. [7] R. H. Fowler, Further studies of Emden's and similar differential equations, Quarterly J. Math, 2 (1931), 259-288. [8] 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. doi: 10.1002/cpa.3160340406. [9] R. A. Johnson, X. Pan and Y. Yi, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302. doi: 10.1016/0362-546X(93)90132-C. [10] R. A. Johnson, X. Pan and Y. Yi, Positive solutions of super-critical elliptic equations and asymptotics, Comm. Partial Differential Equations, 18 (1993), 977-1019. doi: 10.1080/03605309308820958. [11] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269. doi: 10.1007/BF00250508. [12] 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. doi: 10.1007/s002220050285. [13] 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. doi: 10.1016/0022-0396(92)90034-K. [14] Y. Li and W. M. Ni, On conformal scalar curvature equation in $\mathbb{R}^{N}$, Duke Math. J., 57 (1988), 895-924. doi: 10.1215/S0012-7094-88-05740-7. [15] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. [16] R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke Math. J., 99 (1999), 353-418. doi: 10.1215/S0012-7094-99-09913-1. [17] W. M. Ni, On the elliptic equation $\Delta u +K(x)u^(n+2)/(n-2)=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. [18] W. M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math., 36 (1986), 379-399. doi: 10.1002/cpa.3160390306. [19] W. M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. doi: 10.1007/BF03167899. [20] E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|)u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 124 (1993), 239-259. doi: 10.1007/BF00953068.
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