Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball

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• 作者：Yasuhito Miyamoto
• 关键词：Bifurcation diagram ; Exponential growth ; Intersection number ; Elliptic Dirichlet problem ; Singular solution ; 35B32 ; 35J61 ; 35B33 ; 35J25
• 刊名：Annali di Matematica Pura ed Applicata
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：194
• 期：4
• 页码：931-952
• 全文大小：580 KB
• 参考文献：1.Bae, S., Ni, W.-M.: Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on \({\mathbb{R}}^n\) . Math. Ann. 320, 191鈥?10 (2001)MATH MathSciNet View Article
  2.Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437鈥?77 (1983)MATH MathSciNet View Article
  3.Brezis, H., V谩zquez, J.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madr. 10, 443鈥?69 (1997)MATH
  4.Budd, C., Norbury, J.: Semilinear elliptic equations and supercritical growth. J. Differ. Equ. 68, 169鈥?97 (1987)MATH MathSciNet View Article
  5.Crandall, M., Rabinowitz, P.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207鈥?18 (1975)MATH MathSciNet View Article
  6.Dancer, E.: Infinitely many turning points for some supercritical problems. Ann. Mat. Pura Appl. 178, 225鈥?33 (2000)MATH MathSciNet View Article
  7.Dolbeault, J., Flores, I.: Geometry of phase space and solutions of semilinear elliptic equations in a ball. Trans. Am. Math. Soc. 359, 4073鈥?087 (2007)MATH MathSciNet View Article
  8.Gel鈥檉and, I.: Some problems in the theory of quasilinear equations. Am. Math. Soc. Transl. 29, 295鈥?81 (1963)MATH
  9.Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209鈥?43 (1979)MATH MathSciNet View Article
  10.Gui, C.: On positive entire solutions of the elliptic equation \(\Delta u+K(x)u^p=0\) and its applications to Riemannian geometry. Proc. R. Soc. Edinb. A 126, 225鈥?37 (1996)MATH MathSciNet View Article
  11.Gui, C., Ni, W.-M., Wang, X.: On the stability and instability of positive steady states of a semilinear heat equation in \({\mathbb{R}}^n\) . Commun. Pure Appl. Math. 45, 1153鈥?181 (1992)MATH MathSciNet View Article
  12.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鈥?799 (2011)MATH MathSciNet View Article
  13.Jacobsen, J., Schmitt, K.: The Liouville鈥揃ratu鈥揋elfand problem for radial operators. J. Differ. Equ. 184, 283鈥?98 (2002)MATH MathSciNet View Article
  14.Joseph, D., Lundgren, S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241鈥?69 (1972/73)MATH MathSciNet
  15.Korman, P.: Solution curves for semilinear equations on a ball. Proc. Am. Math. Soc. 125, 1997鈥?005 (1997)MATH MathSciNet View Article
  16.Merle, F., Peletier, L.: Positive solutions of elliptic equations involving supercritical growth. Proc. R. Soc. Edinb. A 118, 49鈥?2 (1991)MATH MathSciNet View Article
  17.Miyamoto, Y.: Structure of the Positive Solutions for Supercritical Elliptic Equations in a Ball. J. Math. Pures Appl. 2013 (to appear)
  18.Nagasaki, K., Suzuki, T.: Spectral and related properties about the Emden-Fowler equation \(-\Delta u=\lambda e^u\) on circular domains. Math. Ann. 299, 1鈥?5 (1994)MATH MathSciNet View Article
  19.Suzuki, T.: Semilinear elliptic equations. GAKUTO International Series. Mathematical Sciences and Applications, 3, pp. vi+337. Gakkotosho Co., Ltd, Tokyo (1994). ISBN: 4-7625-0412-2
  
• 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聽, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1618-1891

文摘

Let \(B\subset \mathbb {R}^N\), \(N\ge 3\), be the unit ball. We study the global bifurcation diagram of the solutions of $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\lambda f(u)=0 &{}\quad \text {in}\ B,\\ u=0 &{} \quad \text {on}\ \partial B,\\ u>0 &{} \quad \text {in}\ B, \end{array}\right. } \end{aligned}$$