Browse > Article
http://dx.doi.org/10.4134/JKMS.2011.48.2.431

ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS IN ℝn  

Lai, Baishun (INSTITUTE OF CONTEMPORARY MATHEMATICS HENAN UNIVERSITY, SCHOOL OF MATHEMATICS HENAN UNIVERSITY)
Luo, Qing (SCHOOL OF MATHEMATICS HENAN UNIVERSITY)
Zhou, Shuqing (SCHOOL OF MATHEMATICS AND COMPUTER SCIENCE HUNAN NORMAL UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.2, 2011 , pp. 431-447 More about this Journal
Abstract
We investigate the asymptotic behavior of positive solutions to the elliptic equation (0.1) ${\Delta}u+|x|^{l_1}u^p+|x|^{l_2}u^q=0$ in $\mathbb{R}^n$. We obtain a conclusion that, for n $\geq$ 3, -2 < $l_2$ < $l_1$ $\leq$ 0 and q > p > 1, any positive radial solution to (0.1) has the following properties: $lim_{r{\rightarrow}{\infty}}r^{\frac{2+l_1}{p-1}}\;u$ and $lim_{r{\rightarrow}0}r^{\frac{2+l_2}{q-1}}\;u$ always exist if $\frac{n+1_1}{n-2}$ < p < q, $p\;{\neq}\;\frac{n+2+2l_1}{n-2}$, $q\;{\neq}\;\frac{n+2+2l_2}{n-2}$. In addition, we prove that the singular positive solution of (0.1) is unique under some conditions.
Keywords
semilinear elliptic equation; positive solutions; asymptotic behavior; singular solutions;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta$u + K(lxl)$u^{p}$ = 0 in $R^{n}$, Arch. Rational Mech. Anal. 124 (1993), no. 3, 239-259.   DOI
2 D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972), 241-269.
3 T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc. 333 (1992), no. 1, 365-378.   DOI   ScienceOn
4 Y. Li, Remarks on a semilinear elliptic equation on $R^{n}$, J. Differential Equations 74 (1988), no. 1, 34-49.   DOI
5 Y. Li, Asymptotic behavior of positive solutions of equation $\Delta$u+K(x)$u^{p}$ = 0 in $R^{n}$, J. Differential Equations 95 (1992), no. 2, 304-330.   DOI
6 Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^{n}$, Duke Math. J. 57 (1988), no. 3, 895-924.   DOI
7 Y. Liu, Y. Li, and Y.-B. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations 163 (2000), no. 2, 381-406.   DOI   ScienceOn
8 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), no. 4, 493-529.   DOI
9 W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math. 5 (1988), no. 1, 1-32.   DOI
10 J. Serrin and H. Zou, Classification of positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 1-25.   DOI
11 X.-F. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), no. 2, 549-590.   DOI   ScienceOn
12 F. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), no. 1-2, 29-40.   DOI
13 S. Bae and W.-M. Ni, Existence and infinite multiplicity for an inhomogeneous semi- linear elliptic equation on $R^{n}$, Math. Ann. 320 (2001), no. 1, 191-210.   DOI
14 R. Bamon, I. Flores, and M. del Pino, Ground states of semilinear elliptic equations: a geometric approach, Ann. Inst. H. Poincare Anal. Non Lineaire 17 (2000), no. 5, 551-581.   DOI   ScienceOn
15 G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations 125 (1996), no. 1, 184-214.   DOI   ScienceOn
16 Y.-B. Deng, Y. Li, and Y. Liu, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal. 54 (2003), no. 2, 291-318.   DOI   ScienceOn
17 Y.-B. Deng, Y. Li, and F. Yang, On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem, J. Differential Equations 228 (2006), no. 2, 507-529.   DOI   ScienceOn
18 C.-F. Gui, Positive entire solutions of the equation $\Delta$u + f(x, u) = 0, J. Differential Equations 99 (1992), no. 2, 245-280.   DOI
19 C.-F. Gui, W.-M. Ni, and X.-F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^{n}$, Comm. Pure Appl. Math. 45 (1992), no. 9, 1153-1181.   DOI
20 C.-F. Gui, On positive entire solutions of the elliptic equation $\Delta$u + K(x)$u^{p}$ = 0 and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 2, 225-237.   DOI
21 C.-F. Gui, W.-M. Ni, and X.-F. Wang, Further study on a nonlinear heat equation, J. Differential Equations 169 (2001), no. 2, 588-613.   DOI   ScienceOn
22 K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503-505.   DOI
23 S. Bae, T.-K. Chang, and D.-H. Pank, Infinite multiplicity of positive entire solutions for a semilinear elliptic equation, J. Differential Equations 181 (2002), no. 2, 367-387.   DOI   ScienceOn
24 S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient, J. Differential Equations 237 (2007), no. 1, 159-197.   DOI   ScienceOn
25 S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^{n}$, J. Differential Equations 194 (2003), no. 2, 460-499.   DOI   ScienceOn