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 & Continuous Dynamical Systems - A, 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. doi: 10.1016/j.jde.2007.03.003. Google Scholar

[2]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 460. doi: 10.1016/S0022-0396(03)00172-4. Google Scholar

[3]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$,, J. Differential Equations, 185 (2002), 225. doi: 10.1006/jdeq.2001.4162. Google Scholar

[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. doi: 10.1002/cpa.3160420304. Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[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. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar

[7]

R. H. Fowler, Further studies of Emden's and similar differential equations,, Quarterly J. Math, 2 (1931), 259. Google Scholar

[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. doi: 10.1002/cpa.3160340406. Google Scholar

[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. doi: 10.1016/0362-546X(93)90132-C. Google Scholar

[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. doi: 10.1080/03605309308820958. Google Scholar

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241. doi: 10.1007/BF00250508. Google Scholar

[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. doi: 10.1007/s002220050285. Google Scholar

[13]

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. doi: 10.1016/0022-0396(92)90034-K. Google Scholar

[14]

Y. Li and W. M. Ni, On conformal scalar curvature equation in $\mathbbR^n$,, Duke Math. J., 57 (1988), 895. doi: 10.1215/S0012-7094-88-05740-7. Google Scholar

[15]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381. doi: 10.1006/jdeq.1999.3735. Google Scholar

[16]

R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities,, Duke Math. J., 99 (1999), 353. doi: 10.1215/S0012-7094-99-09913-1. Google Scholar

[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. doi: 10.1512/iumj.1982.31.31040. Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations,, Comm. Pure Appl. Math., 36 (1986), 379. doi: 10.1002/cpa.3160390306. Google Scholar

[19]

W. M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1. doi: 10.1007/BF03167899. Google Scholar

[20]

E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|)u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 124 (1993), 239. doi: 10.1007/BF00953068. Google Scholar

show all references

References:
[1]

S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient,, J. Differential Equations, 237 (2007), 159. doi: 10.1016/j.jde.2007.03.003. Google Scholar

[2]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 460. doi: 10.1016/S0022-0396(03)00172-4. Google Scholar

[3]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $\mathbbR^n$,, J. Differential Equations, 185 (2002), 225. doi: 10.1006/jdeq.2001.4162. Google Scholar

[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. doi: 10.1002/cpa.3160420304. Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[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. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar

[7]

R. H. Fowler, Further studies of Emden's and similar differential equations,, Quarterly J. Math, 2 (1931), 259. Google Scholar

[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. doi: 10.1002/cpa.3160340406. Google Scholar

[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. doi: 10.1016/0362-546X(93)90132-C. Google Scholar

[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. doi: 10.1080/03605309308820958. Google Scholar

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241. doi: 10.1007/BF00250508. Google Scholar

[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. doi: 10.1007/s002220050285. Google Scholar

[13]

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. doi: 10.1016/0022-0396(92)90034-K. Google Scholar

[14]

Y. Li and W. M. Ni, On conformal scalar curvature equation in $\mathbbR^n$,, Duke Math. J., 57 (1988), 895. doi: 10.1215/S0012-7094-88-05740-7. Google Scholar

[15]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381. doi: 10.1006/jdeq.1999.3735. Google Scholar

[16]

R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities,, Duke Math. J., 99 (1999), 353. doi: 10.1215/S0012-7094-99-09913-1. Google Scholar

[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. doi: 10.1512/iumj.1982.31.31040. Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations,, Comm. Pure Appl. Math., 36 (1986), 379. doi: 10.1002/cpa.3160390306. Google Scholar

[19]

W. M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1. doi: 10.1007/BF03167899. Google Scholar

[20]

E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|)u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 124 (1993), 239. doi: 10.1007/BF00953068. Google Scholar

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