Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations

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• 作者：Yasuhito Miyamoto
• 关键词：Quasilinear elliptic equation ; Radial solutions ; Singular solutions ; Intersection number
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：23
• 期：2
• 全文大小：731 KB
• 参考文献：1.Bae S.: Separation structure of positive radial solutions of a semilinear elliptic equation in \({{\mathbf{R}^{\rm n}}}\) . J. Differ. Equ. 194, 460–499 (2003)MathSciNet CrossRef MATH
  2.Bae S.: Infinitely multiplicity and separation structure of positive solutions for a semilinear elliptic equation in \({{\mathbf{R}^{\rm n}}}\) . J. Differ. Equ. 200, 274–311 (2004)CrossRef MATH
  3.Bae S.: Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient. J. Differ. Equ. 237, 159–197 (2007)MathSciNet CrossRef MATH
  4.Bae S., Chang T.-K.: On a class of semilinear elliptic equations in \({{\mathbf{R}^{n}}}\) . J. Differ. Equ. 185, 225–250 (2002)MathSciNet CrossRef MATH
  5.Bidaut-Véron M.-F.: Local and global behavior of solutions of quasilinear equations of Emden–Fowler type. Arch. Ration. Mech. Anal. 107, 293–324 (1989)MathSciNet CrossRef MATH
  6.Cabré X., Sanchón M.: Semi-stable and extremal solutions of reaction equations involving the p-Laplacian. Commun. Pure Appl. Anal. 6, 43–67 (2007)MathSciNet MATH
  7.Clément P., de Figueiredo D., Mitidieri E.: Quasilinear elliptic equations with critical exponents. Topol. Methods Nonlinear Anal. 7, 133–170 (1996)MathSciNet MATH
  8.Egnell H.: Elliptic boundary value problems with singular coefficients and critical nonlinearities. Indiana Univ. Math. J. 38, 235–251 (1989)MathSciNet CrossRef MATH
  9.Emden, R.: Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme (German) In: Teubner, B.G. (ed.) Leipzig; Berlin (1907). https://​en.​wikipedia.​org/​wiki/​Springer_​Vieweg_​Verlag
  10.Ferrero A.: On the solutions of quasilinear elliptic equations with a polynomial-type reaction term. Adv. Differ. Equ. 9, 1201–1234 (2004)MathSciNet MATH
  11.Fila M., King J., Winkler M., Yanagida E.: Optimal lower bound of the grow-up rate for a supercritical parabolic equation. J. Differ. Equ. 228, 339–356 (2006)MathSciNet CrossRef MATH
  12.Fila M., Winkler M., Yanagida E.: Convergence rate for a parabolic equation with supercritical nonlinearity. J. Dyn. Differ. Equ. 17, 249–269 (2005)MathSciNet CrossRef MATH
  13.Fila M., Winkler M., Yanagida E.: Slow convergence to zero for a parabolic equation with a supercritical nonlinearity. Math. Ann. 340, 477–496 (2008)MathSciNet CrossRef MATH
  14.Franca M.: Classification of positive solutions of p–Laplace equation with a growth term. Arch. Math. (Brno) 40, 415–434 (2005)MathSciNet MATH
  15.Franca M.: Some results on the m-Laplace equations with two growth terms. J. Dyn. Differ. Equ. 17, 391–425 (2005)MathSciNet CrossRef MATH
  16.Franca M.: Fowler transformation and radial solutions for quasilinear elliptic equations. I. The subcritical and the supercritical case. Can. Appl. Math. Q. 16, 123–159 (2008)MathSciNet MATH
  17.Franca M.: Fowler transformation and radial solutions for quasilinear elliptic equations. II. Nonlinearities of mixed type. Ann. Mat. Pura Appl. 189, 67–94 (2010)MathSciNet CrossRef MATH
  18.Gui C., Ni W.-M., Wang X.: On the stability and instability of positive steady states of a semilinear heat equation in \({{\mathbb{R}^{n}}}\) . Comm. Pure Appl. Math. 45, 1153–1181 (1992)MathSciNet CrossRef MATH
  19.Gui C., Ni W.-M., Wang X.: Further study on a nonlinear heat equation. J. Differ. Equ. 169, 588–613 (2001)MathSciNet CrossRef MATH
  20.Guo Z., Wei J.: Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent. Trans. Am. Math. Soc. 363, 4777–4799 (2011)MathSciNet CrossRef MATH
  21.Jacobsen J., Schmitt K.: The Liouville–Bratu–Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)MathSciNet CrossRef MATH
  22.Joseph, Y., Lundgren, T.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1972/73)
  23.Liu Y., Li Y., Deng Y.: Separation property of solutions for a semilinear elliptic equation. J. Differ. Equ. 163, 381–406 (2000)MathSciNet CrossRef MATH
  24.Matano H., Merle F.: On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math. 57, 1494–1541 (2004)MathSciNet CrossRef MATH
  25.Matano H., Merle F.: Classification of type I and type II behaviors for a supercritical nonlinear heat equation. J. Funct. Anal. 256, 992–1064 (2009)MathSciNet CrossRef MATH
  26.Matano H., Merle F.: Threshold and generic type I behaviors for a supercritical nonlinear heat equation. J. Funct. Anal. 261, 716–748 (2011)MathSciNet CrossRef MATH
  27.Miyamoto Y.: Structure of the positive solutions for supercritical elliptic equations in a ball. J. Math. Pures Appl. 102, 672–701 (2014)MathSciNet CrossRef MATH
  28.Mizoguchi N., Yanagida E.: Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation. Math. Ann. 307, 663–675 (1997)MathSciNet CrossRef MATH
  29.Mizoguchi N.: Multiple blowup of solutions for a semilinear heat equation. Math. Ann. 331, 461–473 (2005)MathSciNet CrossRef MATH
  30.Mizoguchi N.: Rate of type II blowup for a semilinear heat equation. Math. Ann. 339, 839–877 (2007)MathSciNet CrossRef MATH
  31.Poláčik P., Yanagida E.: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003)MathSciNet CrossRef MATH
  32.Poláčik P., Yanagida E.: Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation. J. Dyn. Differe. Equ. 21, 329–341 (2009)MathSciNet CrossRef MATH
  33.Tso K.: Remarks on critical exponents for Hessian operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 113–122 (1990)MathSciNet MATH
  34.Wang X.-F.: On the Cauchy problem for reaction-diffusion equations. Trans. Am. Math. Soc. 337, 549–590 (1993)MathSciNet CrossRef MATH
  
• 作者单位：Yasuhito Miyamoto (1)
  
  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-9004

文摘

We study the positive solution \({u(r,\rho)}\) of the quasilinear elliptic equation $$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime}+|u|^{p-1}u=0, & 0 < r < \infty,\\ u(0) = \rho > 0,\ u^{\prime}(0)=0.\end{cases}$$This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u *(r) under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and p. A generalized Joseph–Lundgren exponent, which we denote by \({p^*_{JL}}\), is obtained. We study the intersection numbers between \({u(r,\rho)}\) and u *(r) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\), and see that \({p^*_{JL}}\) plays an important role. We also determine the bifurcation diagram of the problem $$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime} + \lambda(u+1)^p=0, & 0 < r < 1,\\ u(r) > 0, & 0 \le r < 1,\\ u^{\prime}(0)=0,\ u(1)=0.\end{cases}$$The main technique used in the proofs is a phase plane analysis. Mathematics Subject Classification 35B05 35B32 35C10 70K05 Keywords Quasilinear elliptic equation Radial solutions Singular solutions Intersection number This work was supported by JSPS KAKENHI Grant Number 24740100.