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

Soohyun Bae

doi:10.3934/dcds.2013.33.555

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

On the elliptic equation Δu+K u^p = 0 in $\mathbb{R}$ⁿ

Soohyun Bae ^1,

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

Received July 2011 Revised April 2012 Published September 2012

Abstract
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This paper deals with the elliptic equation Δu+K(|x|) u^p = 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 $\infty$.

Keywords: singular solution, fast decay, slow decay, Semilinear elliptic equation, asymptotically self-similar solution, phase-plane analysis, Delaunay-Fowler-type solution., positive solution.

Mathematics Subject Classification: Primary: 35J61, 35B40, 35C06; Secondary: 70K0.

Citation: Soohyun Bae. On the elliptic equation Δu+K u^p = 0 in $\mathbb{R}$ⁿ. Discrete & 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. Google Scholar
[2]	S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$, J. Differential Equations, 194 (2003), 460-499. 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), 225-250. 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), 271-297. 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), 615-622. 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), 485-506. 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), 259-288. 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), 525-598. 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), 1279-1302. 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), 977-1019. 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), 241-269. 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), 233-272. 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), 304-330. 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), 895-924. 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), 381-406. 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), 353-418. 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), 493-529. 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), 379-399. 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), 1-32. 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), 239-259. doi: 10.1007/BF00953068. Google Scholar

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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. Google Scholar
[2]	S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbbR^n$, J. Differential Equations, 194 (2003), 460-499. 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), 225-250. 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), 271-297. 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), 615-622. 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), 485-506. 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), 259-288. 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), 525-598. 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), 1279-1302. 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), 977-1019. 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), 241-269. 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), 233-272. 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), 304-330. 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), 895-924. 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), 381-406. 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), 353-418. 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), 493-529. 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), 379-399. 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), 1-32. 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), 239-259. doi: 10.1007/BF00953068. Google Scholar

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On the elliptic equation Δu+K u^p = 0 in $\mathbb{R}$ⁿ

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