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Mathematische Annalen

, Volume 361, Issue 3–4, pp 787–809 | Cite as

Nonradial maximizers for a Hénon type problem and symmetry breaking bifurcations for a Liouville-Gel’fand problem with a vanishing coefficient

  • Yasuhito MiyamotoEmail author
Article

Abstract

We study the radiality and nonradiality of the maximizers of
$$\begin{aligned} \sup \left\{ \int _D|x|^{\alpha }(e^u-1)dx;\ u\in H_0^1(D),\ \int _D\left| \nabla u\right| ^2dx=\beta ^2\right\} \end{aligned}$$
from a viewpoint of the bifurcation diagram of the Euler-Lagrange equation
$$\begin{aligned} \Delta u+\beta ^2\frac{|x|^{\alpha }e^u}{\int _D|x|^{\alpha }ue^udx}=0\ \ \hbox {in}\ \ D,\qquad u=0\ \ \hbox {on}\ \ \partial D, \end{aligned}$$
where \(D:=\{x\in {\mathbb {R}}^2;\ |x|<1\}\). We show that the maximizers are nonradial if
$$\begin{aligned} \beta >\sqrt{8\pi \left\{ (\alpha +2)\log \frac{2\alpha +4}{\alpha } -\frac{\alpha +4}{2}\right\} } \end{aligned}$$
and that the maximizers are radial if \(\alpha >0\) or \(\beta >0\) is small. We also give the bifurcation diagrams of the Liouville-Gel’fand problem with a vanishing coefficient
$$\begin{aligned} \Delta u+\lambda |x|^{\alpha }e^u=0\ \ \hbox {in}\ \ D,\qquad u=0\ \ \hbox {on}\ \ \partial D \end{aligned}$$
and of the associated mean field equation. The analysis is based on exact solutions and explicit calculations.

Mathematics Subject Classification

Primary 35B32 35J20 Secondary 35J61 35B09 

References

  1. 1.
    Bonheure, D., Serra, E., Tarallo, M.: Symmetry of extremal functions in Moser-Trudinger inequalities and a Hénon type problem in dimension two. Adv. Differ. Equ. 13, 105–138 (2008)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^u\) in two dimensions. Comm. Partial Differ. Equ. 16, 1223–1253 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Calanchi, M., Terraneo, E.: Non-radial maximizers for functionals with exponential non-linearity in \({\mathbb{R}}^2\). Adv. Nonlinear Stud. 5, 337–350 (2005)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, C., Lin, C.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56, 1667–1727 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, G., Zhou, J., Ni, W.-M.: Algorithms and visualization for solutions of nonlinear elliptic equations. Internat J. Bifur. Chaos Appl. Sci. Eng. 10, 1565–1612 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Crandall, M., Rabinowitz, P.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dancer, E.: Global breaking of symmetry of positive solutions on two-dimensional annuli. Differ. Integral Equ. 5, 903–913 (1992)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gazzini, M., Serra, E.: The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 281–302 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gel’fand, I.: Some problems in the theory of quasilinear equations. Amer. Math. Soc. Transl. 29, 295–381 (1963)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Golubitsky, M., Stewart, I., Schaeffer, D.: Singularities and groups in bifurcation theory, vol. II. In: Applied Mathematical Sciences, vol. 69. Springer, New York, pp xvi+533 (1988)Google Scholar
  11. 11.
    Gurtin, M., Matano, H.: On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46, 301–317 (1988)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hartman, P., Wintner, A.: On the local behavior of solutions of non-parabolic partial differential equations. Amer. J. Math. 75, 449–476 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Helffer, B., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Owen, M.: Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains. Comm. Math. Phys. 202, 629–649 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Holzmann, M., Kielhöfer, H.: Uniqueness of global positive solution branches of nonlinear elliptic problems. Math. Ann. 300, 221–241 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Joseph, D., Lundgren, T.: Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49, 241–269 (1972/73)Google Scholar
  16. 16.
    Kawohl, B.: Rearrangements and convexity of level sets in PDE. In: Lecture Notes in Mathematics, vol. 1150. Springer, Berlin, iv+136 pp (1985)Google Scholar
  17. 17.
    Li, Y., Shafrir, I.: Blow-up analysis for solutions of \(-\Delta u=Ve^u\) in dimension two. Indiana Univ. Math. J. 43, 1255–1270 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lin, S.: On non-radially symmetric bifurcation in the annulus. J. Differ. Equ. 80, 251–279 (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)Google Scholar
  20. 20.
    Nagasaki, K., Suzuki, T.: Spectral and related properties about the Emden-Fowler equation \(-\Delta u=\lambda e^u\) on circular domains. Math. Ann. 299, 1–15 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Sattinger, D.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1971/72)Google Scholar
  23. 23.
    Secchi, S., Serra, E.: Symmetry breaking results for problems with exponential growth in the unit disk. Commun. Contemp. Math. 8, 823–839 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sherman, J., Morrison, W.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat. 21, 124–127 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Smets, D., Willem, M., Su, J.: Non-radial ground states for the Hénon equation. Commun. Contemp. Math. 4, 467–480 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Suzuki, T.: Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 367–397 (1992)zbMATHGoogle Scholar
  27. 27.
    Suzuki, T.: Semilinear elliptic equations. In: GAKUTO International Series. Mathematical Sciences and Applications, vol. 3. Gakkotosho Co., Ltd, Tokyo, pp vi+337. ISBN: 4-7625-0412-2 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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