Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities

Karali, Georgia ; Sourdis, Christos

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 131-170 / Harvested from Numdam

Résumé

We consider the singular perturbation problem $ -{\epsilon }^{2}\mathrm{\Delta }u+(u-a(|x|))(u-b(|x|))=0$ in the unit ball of $ {\mathbb{R}}^{N}$, $ N⩾1$, under Neumann boundary conditions. The assumption that $ a(r)-b(r)$ 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 $ \epsilon \to 0$. The solution $ {u}_{+}$ is asymptotically stable, whereas $ {u}_{-}$ has Morse index one, in the radial class. If $ N⩾2$, we prove that the Morse index of $ {u}_{-}$, in the general class, is asymptotically given by $ [c+o(1)]{\epsilon }^{-\frac{2}{3}(N-1)}$ as $ \epsilon \to 0$, with $ c>0$ a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of $ {\epsilon }_{k}>0$, with $ {\epsilon }_{k}\to 0$ as $ k\to +\infty $, such that non-radial solutions bifurcate from the unstable branch $ \{({u}_{-}(\epsilon ),\epsilon ),\,\epsilon >0\}$ at $ \epsilon ={\epsilon }_{k}$, $ k=1,2,\dots $. 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.

Détail

Publié le : 2012-01-01

Zbl 1242.35114

DOI : https://doi.org/10.1016/j.anihpc.2011.09.005
Classification:  35J25,  35J20,  35B33,  35B40

Bibliographie

[1] N.D. Alikakos, P.W. Bates, J.W. Cahn, P.C. Fife, G. Fusco, G.B. Tanoglu, Analysis of the corner layer problem in anisotropy, Discrete Contin. Dyn. Syst. 6 (2006), 237-255 | Zbl 1235.74011

[2] G. Allain, A. Beaulieu, Singly periodic solutions of a semilinear equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1277-1297 | Numdam | Zbl 1172.35375

[3] A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I, Comm. Math. Phys. 235 (2003), 427-466 | Zbl 1072.35019

[4] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York (1999) | Zbl 0938.34001

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

[6] V.F. Butuzov, 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 1210.35147

[7] V.F. Butuzov, N.N. Nefedov, K.R. Schneider, Singularly perturbed elliptic problems in the case of exchange of stabilities, J. Differential Equations 169 (2001), 373-395 | Zbl 0974.35039

[8] L.A. Caffarelli, J.-M. Roquejoffre, 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 1189.35084

[9] R.S. Cantrell, C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, Wiley Ser. Math. Comput. Biol. (2003) | Zbl 1059.92051

[10] R. Carmona, B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, Comm. Math. Phys. 80 (1981), 59-98 | Zbl 0464.35085

[11] P. Chossat, R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, Adv. Ser. Nonlinear Dynam. vol. 15, World Scientific (2000) | Zbl 0968.37001

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

[13] M. Conti, S. Terracini, G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math. 195 (2005), 524-560 | Zbl 1126.35016

[14] M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Ration. Mech. Anal. 52 (1973), 161-180 | Zbl 0275.47044

[15] E.N. Dancer, New solutions of equations on $ {\mathbb{R}}^{n}$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) XXX (2001), 535-563 | Numdam | Zbl 1025.35009

[16] E.N. Dancer, Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), 434-475 | Zbl 0815.35024

[17] E.N. Dancer, S. Yan, On the superlinear Lazer–McKenna conjecture, J. Differential Equations 210 (2005), 317-351 | Zbl 1190.35082

[18] M. Del Pino, M. Kowalczyk, J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math. 60 (2007), 113-146 | Zbl 1123.35003

[19] Y. Du, K. Nakashima, Morse index of layered solutions to the heterogeneous Allen–Cahn equation, J. Differential Equations 238 (2007), 87-117 | Zbl 1121.35042

[20] F. Dumortier, R. Roussarie, 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 1007.34058

[21] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53-98 | Zbl 0476.34034

[22] P.C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Ration. Mech. Anal. 52 (1973), 205-232 | Zbl 0268.35007

[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] G. Fusco, C. Pignotti, Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators, J. Differential Equations 244 (2008), 514-554 | Zbl 1142.34057

[25] C. Gallo, D. Pelinovsky, On the Thomas–Fermi ground state in a harmonic potential, Asymptot. Anal. 73 no. 1 (2011), 53-96 | Zbl 1225.35217

[26] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243 | Zbl 0425.35020

[27] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York (1983) | Zbl 0691.35001

[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 2764908

[29] M. Golubitsky, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. I, Appl. Math. Sci. vol. 51, Springer-Verlag, New York (1985) | MR 771477

[30] M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. II, Appl. Math. Sci. vol. 69, Springer-Verlag, New York (1988) | MR 950168

[31] M. Grossi, Radial solutions for the Brezis–Nirenberg problem involving large nonlinearities, J. Funct. Anal. 254 (2008), 2995-3036 | MR 2418617 | Zbl 1154.35050

[32] M. Hǎrǎguş-Courcelle, R.L. Pego, Travelling waves of the KP equations with transverse modulations, C. R. Acad. Sci. Paris 328 (1999), 227-232 | MR 1674527 | Zbl 0924.35142

[33] S.P. Hastings, J.B. Mcleod, 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 555581 | Zbl 0426.34019

[34] P. Hislop, I.M. Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators, Appl. Math. Sci. vol. 113, Springer-Verlag, New York (1996) | MR 1361167 | Zbl 0855.47002

[35] R. Ignat, V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate, J. Funct. Anal. 233 (2006), 260-306 | MR 2204681 | Zbl 1106.58009

[36] M. Iida, K. Nakashima, E. Yanagida, 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 2387255 | Zbl 1141.35319

[37] C.K.R.T. Jones, Geometric singular perturbation theory, Dynamical Systems, Montecatini Terme, 1994, Lecture Notes in Math. vol. 1609, Springer, Berlin (1995), 44-118 | MR 1374108 | Zbl 0840.58040

[38] Y. Kabeya, K. Tanaka, 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 1683050 | Zbl 0930.35064

[39] T.J. Kaper, 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 1718893

[40] H. Kielhöfer, A bifurcation theorem for potential operators, J. Funct. Anal. 77 (1988), 1-8 | MR 930387 | Zbl 0643.47057

[41] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Appl. Math. Sci. vol. 156, Springer-Verlag, New York (2004) | MR 2004250 | Zbl 1032.35001

[42] M. Krupa, P. Szmolyan, 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 1846574 | Zbl 1013.34057

[43] S.S. Lin, On non-radially symmetric bifurcation in the annulus, J. Differential Equations 80 (1989), 251-279 | MR 1011150 | Zbl 0688.35005

[44] S. Lorca, M. Montenegro, Spike solutions of a nonlinear Schrödinger equation with degenerate potential, J. Math. Anal. Appl. 295 (2004), 276-286 | MR 2064425 | Zbl 1051.35089

[45] F. Mahmoudi, A. Malchiodi, J. Wei, Transition layer for the heterogeneous Allen–Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 609-631 | Numdam | MR 2422081 | Zbl 1148.35030

[46] F. Mahmoudi, R. Mazzeo, F. Pacard, Constant mean curvature hypersurfaces condensing along a submanifold, Geom. Funct. Anal. 16 (2006), 924-958 | MR 2255386 | Zbl 1108.53031

[47] A. Malchiodi, M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math. 55 (2002), 1507-1568 | MR 1923818 | Zbl 1124.35305

[48] R. Mazzeo, F. Pacard, Constant curvature foliations in asymptotically hyperbolic spaces, Rev. Mat. Iberoamericana 27 no. 1 (2011), 303-333 | MR 2815739 | Zbl 1214.53024

[49] H. Matano, Asymptotic behaviour and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), 401-454 | MR 555661 | Zbl 0445.35063

[50] W.-M. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851 | MR 1115095 | Zbl 0754.35042

[51] F. Pacard, Radial and non-radial solutions of $ -\mathrm{\Delta }u=\lambda f(u)$, on an annulus of $ {\mathbb{R}}^{n}$, $ n⩾3$, J. Differential Equations 101 (1993), 103-138 | MR 1199485 | Zbl 0799.35089

[52] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270-291 | MR 1162728 | Zbl 0763.35087

[53] K. Sakamoto, H. Suzuki, Spherically symmetric internal layers for activator–inhibitor systems II. Stability and symmetry breaking bifurcations, J. Differential Equations 204 (2004), 93-122 | MR 2076160 | Zbl 1068.35014

[54] K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal. 42 (2005), 55-104 | MR 2133874 | Zbl 1210.35019

[55] S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), 185-210 | MR 2094383 | Zbl 1077.35095

[56] S. Schecter, C. Sourdis, Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations, J. Dynam. Differential Equations 22 (2010), 629-655 | MR 2734473 | Zbl 1213.34072

[57] J. Shatah, C. Zeng, Periodic solutions for Hamiltonian systems under strong constraining forces, J. Differential Equations 186 (2002), 572-585 | MR 1942222 | Zbl 1014.37038

[58] J. Smoller, A. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. Math. Phys. 105 (1986), 415-441 | MR 848648 | Zbl 0608.35004

[59] J. Smoller, A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63-95 | MR 1037143 | Zbl 0721.58011

[60] J.A. Smoller, A.G. Wasserman, Symmetry, degeneracy, and uniqueness in semilinear elliptic equations, infinitesimal symmetry-breaking, J. Funct. Anal. 89 (1990), 364-409 | MR 1042215 | Zbl 0702.35016

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

[62] C. Sourdis, P.C. Fife, Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces, Adv. Differential Equations 12 (2007), 623-668 | MR 2319451 | Zbl 1157.34047

[63] M.E. Taylor, Partial Differential Equations II, Qualitative Studies of Linear Equations, Appl. Math. Sci. vol. 116, Springer-Verlag, New York (1996) | MR 1395149

[64] S.-K. Tin, N. Kopell, C.K.R.T. Jones, Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal. 31 (1994), 1558-1576 | MR 1302675 | Zbl 0813.34024

[65] A. Vanderbauwhede, Local Bifurcation and Symmetry, Res. Notes Math. vol. 75, Pitman, Boston (1982) | MR 697724 | Zbl 0539.58022

[66] W. Walter, Ordinary Differential Equations, Grad. Texts in Math. vol. 182, Springer-Verlag, New York (1998) | MR 1629775