Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Miyamoto, Yasuhito

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 59-81

Let $A:=\{a<|x|<1+a\}\subset {ℝ}^N$ and $p⩾2$. We consider the Neumann problem

$${ϵ}^2\Delta u-u+u^p=0\text{in}A,{\partial }_{\nu }u=0\text{on}\partial A.$$

Let $\lambda =1/{ϵ}^2$. When λ is large, we prove the existence of a smooth curve $\left\{\right(\lambda ,u\left(\lambda \right)\left)\right\}$ consisting of radially symmetric and radially decreasing solutions concentrating on $\left\{\right|x|=a\}$. Moreover, checking the transversality condition, we show that this curve has infinitely many symmetry breaking bifurcation points from which continua consisting of nonradially symmetric solutions emanate. If $N=2$, then the closure of each bifurcating continuum is locally homeomorphic to a disk. When the domain is a rectangle $(0,1)\times (0,a)\subset {ℝ}^2$, we show that a curve consisting of one-dimensional solutions concentrating on $\left\{0\right\}\times [0,a]$ has infinitely many symmetry breaking bifurcation points. Extending this solution with even reflection, we obtain a new entire solution.

Classification : 35J91, 35B32, 35B25, 35P15
Keywords: Symmetry breaking bifurcation, Asymptotic transversality, Singular perturbation, Boundary concentration, Nonradially symmetric solutions

[1] A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I, Commun. Math. Phys. 235 (2003), 427-466 | MR | Zbl

[2] A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. II, Indiana Univ. Math. J. 53 (2004), 297-329 | MR | Zbl

[3] T. Bartsch, M. Clapp, M. Grossi, F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann., in press. | MR

[4] M. Crandall, P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340 | MR | Zbl

[5] E.N. Dancer, On the existence of bifurcating solutions in the presence of symmetries, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 321-336 | MR | Zbl

[6] E.N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Differential Integral Equations 5 (1992), 903-913 | MR | Zbl

[7] P. De Groen, G. Karadzhov, Metastability in the shadow system for Gierer–Meinhardtʼs equations, Electron. J. Differential Equations 50 (2002), 22 | MR | EuDML | Zbl

[8] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), 30-39

[9] F. Gladiali, M. Grossi, F. Pacella, P. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations 40 (2011), 295-317 | MR | Zbl

[10] C. Gui, J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), 522-538 | Zbl

[11] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics vol. 840, Springer-Verlag, Berlin, New York (1981) | Zbl

[12] E. Keller, L. Segal, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415 | Zbl

[13] S.-S. Lin, On non-radially symmetric bifurcation in the annulus, J. Differential Equations 80 (1989), 251-279 | Zbl

[14] S.-S. Lin, Positive radial solutions and nonradial bifurcation for semilinear elliptic equations in annular domains, J. Differential Equations 86 (1990), 367-391 | Zbl

[15] S.-S. Lin, Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains, Trans. Amer. Math. Soc. 332 (1992), 775-791 | Zbl

[16] S.-S. Lin, Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. Differential Equations 120 (1995), 255-288 | Zbl

[17] A. Malchiodi, M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Commun. Pure Appl. Math. 55 (2002), 1507-1568 | Zbl

[18] Y. Miyamoto, Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal. 256 (2009), 747-776 | Zbl

[19] Y. Miyamoto, Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry, J. Math. Anal. Appl. 357 (2009), 89-97 | Zbl

[20] Y. Miyamoto, Global branches from the second eigenvalue for a semilinear Neumann problem in a ball, J. Differential Equations 249 (2010), 1853-1870 | Zbl

[21] W.-M. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Commun. Pure Appl. Math. 44 (1991), 819-851 | Zbl

[22] F. Pacard, Radial and nonradial solutions of $-\Delta u=\lambda f\left(u\right)$, on an annulus of ${ℝ}^n$ $n⩾3$, J. Differential Equations 101 (1993), 103-138 | Zbl

[23] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513 | Zbl

[24] J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc. 354 (2002), 3117-3154 | Zbl

[25] J. Smoller, A. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Commun. Math. Phys. 105 (1986), 415-441 | Zbl

[26] J. Smoller, A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63-95 | EuDML | Zbl

[27] P. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 107-112 | EuDML | Zbl | Numdam

[28] T. Wakasa, Exact eigenvalues and eigenfunctions associated with linearization for Chafee–Infante problem, Funkcial. Ekvac. 49 (2006), 321-336 | Zbl

[29] T. Wakasa, S. Yotsutani, Representation formulas for some 1-dimensional linearized eigenvalue problems, Commun. Pure Appl. Anal. 7 (2008), 745-763 | Zbl

[30] J. Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), 104-133 | Zbl

[31] J. Wei, On single interior spike solutions of the Gierer–Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999), 353-378 | Zbl