We consider the singular perturbation problem $-{\u03f5}^{2}\Delta u+(u-a(\left|x\right|\left)\right)(u-b(\left|x\right|\left)\right)=0$ in the unit ball of ${\mathbb{R}}^{N}$, $N\u2a7e1$, under Neumann boundary conditions. The assumption that $a\left(r\right)-b\left(r\right)$ changes sign in $(0,1)$, known as the case of exchange of stabilities, is the main source of difficulty. More precisely, under the assumption that $a-b$ has one simple zero in $(0,1)$, we prove the existence of two radial solutions ${u}_{+}$ and ${u}_{-}$ that converge uniformly to $\mathrm{max}\{a,b\}$, as $\u03f5\to 0$. The solution ${u}_{+}$ is asymptotically stable, whereas ${u}_{-}$ has Morse index one, in the radial class. If $N\u2a7e2$, we prove that the Morse index of ${u}_{-}$, in the general class, is asymptotically given by $[c+o\left(1\right)]{\u03f5}^{-\frac{2}{3}(N-1)}$ as $\u03f5\to 0$, with $c>0$ a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of ${\u03f5}_{k}>0$, with ${\u03f5}_{k}\to 0$ as $k\to +\infty $, such that non-radial solutions bifurcate from the unstable branch $\{({u}_{-}\left(\u03f5\right),\u03f5),\phantom{\rule{0.166667em}{0ex}}\u03f5>0\}$ at $\u03f5={\u03f5}_{k}$, $k=1,2,\cdots $. Our approach is perturbative, based on the existence and non-degeneracy of solutions of a “limit” problem. Moreover, our method of proof can be generalized to treat, in a unified manner, problems of the same nature where the singular limit is continuous but non-smooth.

Keywords: Corner layer, Exchange of stabilities, Geometric singular perturbation theory, Non-radial bifurcations

@article{AIHPC_2012__29_2_131_0, author = {Karali, Georgia and Sourdis, Christos}, title = {Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {131--170}, publisher = {Elsevier}, volume = {29}, number = {2}, year = {2012}, doi = {10.1016/j.anihpc.2011.09.005}, zbl = {1242.35114}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/} }

TY - JOUR AU - Karali, Georgia AU - Sourdis, Christos TI - Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 131 EP - 170 VL - 29 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/ DO - 10.1016/j.anihpc.2011.09.005 LA - en ID - AIHPC_2012__29_2_131_0 ER -

%0 Journal Article %A Karali, Georgia %A Sourdis, Christos %T Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 131-170 %V 29 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/ %R 10.1016/j.anihpc.2011.09.005 %G en %F AIHPC_2012__29_2_131_0

Karali, Georgia; Sourdis, Christos. Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, pp. 131-170. doi : 10.1016/j.anihpc.2011.09.005. http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.005/

[1] Analysis of the corner layer problem in anisotropy, Discrete Contin. Dyn. Syst. 6 (2006), 237-255 | Zbl

, , , , , ,[2] Singly periodic solutions of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1277-1297 | EuDML | Numdam | Zbl

, ,[3] Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I, Comm. Math. Phys. 235 (2003), 427-466 | Zbl

, , ,[4] Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York (1999) | Zbl

, ,[5] Some applications of the method of sub- and supersolutions, Lecture Notes in Math. vol. 782, Springer, Berlin (1980), 16-41

, ,[6] On the stability and domain of attraction of asymptotically non-smooth stationary solutions to a singularly perturbed parabolic equation, Comput. Math. Math. Phys. 46 (2006), 413-424 | Zbl

,[7] Singularly perturbed elliptic problems in the case of exchange of stabilities, J. Differential Equations 169 (2001), 373-395 | Zbl

, , ,[8] Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal. 183 (2007), 457-487 | Zbl

, ,[9] Spatial Ecology via Reaction–Diffusion Equations, Wiley Ser. Math. Comput. Biol. (2003) | Zbl

, ,[10] Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, Comm. Math. Phys. 80 (1981), 59-98 | Zbl

, ,[11] Methods in Equivariant Bifurcations and Dynamical Systems, Adv. Ser. Nonlinear Dynam. vol. 15, World Scientific (2000) | Zbl

, ,[12] Symmetry breakdown from bifurcation, Lett. Nuovo Cimento 31 (1981), 600-602

,[13] Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math. 195 (2005), 524-560 | Zbl

, , ,[14] Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. Anal. 52 (1973), 161-180 | Zbl

, ,[15] New solutions of equations on ${\mathbb{R}}^{n}$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) XXX (2001), 535-563 | EuDML | Numdam | Zbl

,[16] Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), 434-475 | Zbl

, ,[17] On the superlinear Lazer–McKenna conjecture, J. Differential Equations 210 (2005), 317-351 | Zbl

, ,[18] Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math. 60 (2007), 113-146 | Zbl

, , ,[19] Morse index of layered solutions to the heterogeneous Allen–Cahn equation, J. Differential Equations 238 (2007), 87-117 | Zbl

, ,[20] Geometric singular perturbation theory beyond normal hyperbolicity, Multiple-time-scale Dynamical Systems, Minneapolis, MN, 1997, IMA Vol. Math. Appl. vol. 122, Springer, New York (2001), 29-63 | Zbl

, ,[21] Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53-98 | Zbl

,[22] Semilinear elliptic boundary value problems with small parameters, Arch. Ration. Mech. Anal. 52 (1973), 205-232 | Zbl

,[23] P.C. Fife, A phase plane analysis of a corner layer problem arising in the study of crystalline grain boundaries, unpublished preprint, University of Utah.

[24] Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators, J. Differential Equations 244 (2008), 514-554 | Zbl

, ,[25] On the Thomas–Fermi ground state in a harmonic potential, Asymptot. Anal. 73 no. 1 (2011), 53-96 | Zbl

, ,[26] Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243 | Zbl

, , ,[27] Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York (1983) | Zbl

, ,[28] F. Gladiali, M. Grossi, F. Pacella, P.N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, preprint, 2009. | MR

[29] Singularities and Groups in Bifurcation Theory, vol. I, Appl. Math. Sci. vol. 51, Springer-Verlag, New York (1985) | MR

, ,[30] Singularities and Groups in Bifurcation Theory, vol. II, Appl. Math. Sci. vol. 69, Springer-Verlag, New York (1988) | MR

, , ,[31] Radial solutions for the Brezis–Nirenberg problem involving large nonlinearities, J. Funct. Anal. 254 (2008), 2995-3036 | MR | Zbl

,[32] Travelling waves of the KP equations with transverse modulations, C. R. Acad. Sci. Paris 328 (1999), 227-232 | MR | Zbl

, ,[33] A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation, Arch. Ration. Mech. Anal. 73 (1980), 31-51 | MR | Zbl

, ,[34] Introduction to Spectral Theory with Applications to Schrödinger Operators, Appl. Math. Sci. vol. 113, Springer-Verlag, New York (1996) | MR | Zbl

, ,[35] The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate, J. Funct. Anal. 233 (2006), 260-306 | MR | Zbl

, ,[36] On certain one-dimensional elliptic systems under different growth conditions at respective infinities, Asymptotic Analysis and Singularities, Adv. Stud. Pure Math. vol. 47–2 (2007), 565-572 | MR | Zbl

, , ,[37] Geometric singular perturbation theory, Dynamical Systems, Montecatini Terme, 1994, Lecture Notes in Math. vol. 1609, Springer, Berlin (1995), 44-118 | MR | Zbl

,[38] Uniqueness of positive solutions of semilinear elliptic equations in ${\mathbb{R}}^{N}$ and Séréʼs non-degeneracy condition, Comm. Partial Differential Equations 24 (1999), 563-598 | MR | Zbl

, ,[39] An introduction to geometric methods and dynamical systems theory for singular perturbation problems, Analyzing Multiscale Phenomena Using Singular Perturbation Methods, Baltimore, MD, 1998, Proc. Sympos. Appl. Math. vol. 56, Amer. Math. Soc., Providence, RI (1999), 85-131 | MR

,[40] A bifurcation theorem for potential operators, J. Funct. Anal. 77 (1988), 1-8 | MR | Zbl

,[41] Bifurcation Theory: An Introduction with Applications to PDEs, Appl. Math. Sci. vol. 156, Springer-Verlag, New York (2004) | MR | Zbl

,[42] Geometric analysis of the singularly perturbed planar fold, Multiple-time-scale Dynamical Systems, Minneapolis, MN, 1997, IMA Vol. Math. Appl. vol. 122, Springer, New York (2001), 89-116 | MR | Zbl

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

,[44] Spike solutions of a nonlinear Schrödinger equation with degenerate potential, J. Math. Anal. Appl. 295 (2004), 276-286 | MR | Zbl

, ,[45] Transition layer for the heterogeneous Allen–Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 609-631 | EuDML | Numdam | MR | Zbl

, , ,[46] Constant mean curvature hypersurfaces condensing along a submanifold, Geom. Funct. Anal. 16 (2006), 924-958 | MR | Zbl

, , ,[47] Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math. 55 (2002), 1507-1568 | MR | Zbl

, ,[48] Constant curvature foliations in asymptotically hyperbolic spaces, Rev. Mat. Iberoamericana 27 no. 1 (2011), 303-333 | MR | Zbl

, ,[49] Asymptotic behaviour and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), 401-454 | MR | Zbl

,[50] On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851 | MR | Zbl

, ,[51] Radial and non-radial solutions of $-\Delta u=\lambda f\left(u\right)$, on an annulus of ${\mathbb{R}}^{n}$, $n\u2a7e3$, J. Differential Equations 101 (1993), 103-138 | MR | Zbl

,[52] On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270-291 | MR | Zbl

,[53] Spherically symmetric internal layers for activator–inhibitor systems II. Stability and symmetry breaking bifurcations, J. Differential Equations 204 (2004), 93-122 | MR | Zbl

, ,[54] Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal. 42 (2005), 55-104 | MR | Zbl

,[55] Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), 185-210 | MR | Zbl

,[56] Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations, J. Dynam. Differential Equations 22 (2010), 629-655 | MR | Zbl

, ,[57] Periodic solutions for Hamiltonian systems under strong constraining forces, J. Differential Equations 186 (2002), 572-585 | MR | Zbl

, ,[58] Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. Math. Phys. 105 (1986), 415-441 | MR | Zbl

, ,[59] Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63-95 | EuDML | MR | Zbl

, ,[60] Symmetry, degeneracy, and uniqueness in semilinear elliptic equations, infinitesimal symmetry-breaking, J. Funct. Anal. 89 (1990), 364-409 | MR | Zbl

, ,[61] Sourdis, C., On the equation ${u}^{\u2033}=u({\left|u\right|}^{p}-x)$, $p>1$, preprint, 2011.

[62] Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces, Adv. Differential Equations 12 (2007), 623-668 | MR | Zbl

, ,[63] Partial Differential Equations II, Qualitative Studies of Linear Equations, Appl. Math. Sci. vol. 116, Springer-Verlag, New York (1996) | MR

,[64] Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal. 31 (1994), 1558-1576 | MR | Zbl

, , ,[65] Local Bifurcation and Symmetry, Res. Notes Math. vol. 75, Pitman, Boston (1982) | MR | Zbl

,[66] Ordinary Differential Equations, Grad. Texts in Math. vol. 182, Springer-Verlag, New York (1998) | MR

,*Cited by Sources: *