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Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝn, II

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Abstract

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in ℝn by employing the Li’s method of energy function.

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References

  1. Bae, S.: Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in ℝn. J. Differential Equations, 200, 274–311 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Deng, Y.-B., Li, Y., Liu, Y.: On the stability of the positive radial steady states for a semilinear Cauchy problem. Nonlinear Anal., 54, 291–318 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gui, C.-F.: Positive entire solutions of equation Δu + F(x, u) = 0. J. Differential Equations, 99, 245–280 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Liu, Y., Li, Y., Deng, Y.-B.: Separetion property of solutions for a semilinear elliptic equation. J. Differential Equations, 163, 381–406 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ni, W.-M.: On the elliptic equation \( \Delta u + K\left( x \right)u^{\tfrac{{n + 2}} {{n - 2}}} = 0 \), its generalizations and applications in geometry. Indian Univ. Math. J., 31, 493–529 (1982)

    Article  MATH  Google Scholar 

  6. Ni, W.-M., Yotsutani, S.: Semilinear elliptic equations of Matukuma type and relate topics. Japan J. Appl. Math., 5, 1–32 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Xu, Z.-T.: Oscillation theorems for quasilinear elliptic differential equations. Acta Mathematica Sinica, English Series, 23, 1189–1198 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kang, D.-S.: On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents. Acta Mathematica Sinica, English Series, 25, 435–444 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, Y.: Asymptotic behavior of positive solutions of equation Δu + K(x)u p = 0 in ℝn. J. Differential Equations, 95, 304–330 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gui, C.-F., Ni, W.-M., Wang, X.-F.: On the stability and instability of positive steady state of a semilinear heat equation in ℝn. Comm. Pure Appl. Math., 45, 1153–1181 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, Y., Ni, W.-M.: On conformal scalar curvature equations in ℝn. Duke Math. J., 57, 895–924 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lai, B.-S., Zhou, S.-Q.: Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝn, submitted

  13. Bae, S., Chang, T.-K., Pank, D.-H.: Infinit multiplicity of positive entire solutions for a semilinear elliptic equation. J. Differential Equations, 181, 367–387 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bae, S.: Positive entire solutions of semilinear elliptic equations with quadrtically vanishing coefficient. J. Differential Equations, 237, 159–197 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bae, S.: Separation structure of positive radial solutions of a semilinear elliptic equations in ℝn. J. Differential Equations, 194, 460–499 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fujita, H.: On the blowing up of solutions of the Cauchy problem for u t = Δu + u 1+α. J. Fac. Sci. Univ. Tokyo, 13, 109–124 (1966)

    MATH  Google Scholar 

  17. Gui, C.-F.: On positive entire solutions of the elliptic equation Δu + K(x)u p = 0 and its applications to Riemannian geometry. Proc. Roy. Soc. Edinburgh Sect. A, 126, 225–237 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic equations. Proc. Japan Acad., 49, 503–525 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. Joseph, D. D., Lundgren, T. S.: Quasilinear Dirichlet probems driven by positive sources. Arch. Ration. Mech. Anal., 49, 241–269 (1972)

    MathSciNet  Google Scholar 

  20. Kobyashi, K., Siaro, T., Tanaka, H.: On the growing up problem for semilinear heat equations. J. Math. Soc. Japan, 29(3), 407–424 (1977)

    Article  MathSciNet  Google Scholar 

  21. Lee, T.-Y., Ni, W.-M.: Global existence, Large time behavior and life span of solutions of semilinear parabolic Cauchy problems. Trans. Amer. Math. Soc., 333, 365–371 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Weissler, F.: Existence and nonexistence of global solutuion for semilinear heat equation. Israel J. Math., 38, 29–40 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yanagida, E., Yotsutani, S.: Classification of the structure of positive radial solutions to Δu+K(|x|)u p = 0 in ℝn. Arch. Rational Mech. Anal., 124, 239–259 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Shu Qing Zhou.

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Supported by National Natural Science Foundation of China (Grant Nos. 10901047 and 10971061) and Excellent Youth Program of Hunan Normal University (Grant No. 080640)

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Lai, B.S., Zhou, S.Q. Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝn, II. Acta. Math. Sin.-English Ser. 26, 1723–1738 (2010). https://doi.org/10.1007/s10114-010-8325-y

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  • DOI: https://doi.org/10.1007/s10114-010-8325-y

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