Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, p. 107-126

We consider the equation -ϵ 2 Δu+u=u p in a bounded domain Ω 3 with edges. We impose Neumann boundary conditions, assuming 1<p<5, and prove concentration of solutions at suitable points of ∂Ω on the edges.

@article{AIHPC_2011__28_1_107_0,
     author = {Dipierro, Serena},
     title = {Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     year = {2011},
     pages = {107-126},
     doi = {10.1016/j.anihpc.2010.11.003},
     zbl = {1209.35040},
     mrnumber = {2765513},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_1_107_0}
}
Dipierro, Serena. Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, pp. 107-126. doi : 10.1016/j.anihpc.2010.11.003. http://www.numdam.org/item/AIHPC_2011__28_1_107_0/

[1] R.A. Adams, Sobolev Spaces, Academic Press, New York (1975) | MR 450957 | Zbl 0186.19101

[2] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on n , Progr. Math. vol. 240, Birkhäuser (2005) | MR 2186962

[3] H. Berestycki, P.L. Lions, Nonlinear scalar field equations (part I and part II), Arch. Ration. Mech. Anal. 82 (1983), 313-376 | MR 695535

[4] R.G. Casten, C.J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 no. 2 (1978), 266-273 | Zbl 0338.35055

[5] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York (1984) | MR 768584

[6] E.N. Dancer, J. Wei, On the effect of domain topology in a singular perturbation problem, Topol. Methods Nonlinear Anal. 11 no. 2 (1998), 227-248 | MR 1659466 | Zbl 0926.35015

[7] E.N. Dancer, S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 no. 2 (1999), 241-262 | MR 1696122 | Zbl 0933.35070

[8] M. Del Pino, P. Felmer, J. Wei, On the role of the mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), 63-79 | MR 1742305 | Zbl 0942.35058

[9] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397-408 | MR 867665 | Zbl 0613.35076

[10] J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results, Arch. Ration. Mech. Anal. 196 no. 3 (2010), 907-950 | MR 2644444 | Zbl 1214.35024

[11] J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems. Part II: asymptotic of minimal energy solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 37-56 | Numdam | MR 2580503 | Zbl 1194.35037

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

[13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London (1985) | MR 775683 | Zbl 0695.35060

[14] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia Math. Appl. vol. 61, Cambridge University Press, Cambridge (1996) | MR 1412143 | Zbl 0877.52002

[15] M. Grossi, Some results on a class of nonlinear Schrödinger equations, Math. Z. 235 no. 4 (2000), 687-705 | MR 1801580 | Zbl 0970.35039

[16] M. Grossi, A. Pistoia, J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via non-smooth critical point theory, Calc. Var. Partial Differential Equations 11 no. 2 (2000), 143-175 | MR 1782991 | Zbl 0964.35047

[17] C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 no. 3 (1996), 739-769 | MR 1408543 | Zbl 0866.35039

[18] C. Gui, J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 no. 1 (1999), 1-27 | MR 1721719 | Zbl 1061.35502

[19] C. Gui, J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 no. 3 (2000), 522-538 | MR 1758231 | Zbl 0949.35052

[20] C. Gui, J. Wei, M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 no. 1 (2000), 47-82 | Numdam | MR 1743431 | Zbl 0944.35020

[21] E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Pub. Co. (1955) | MR 64922 | Zbl 0004.21001

[22] M.K. Kwong, Uniqueness of positive solutions of Δu-u+u p =0 in n , Arch. Ration. Mech. Anal. 105 (1989), 243-266 | MR 969899 | Zbl 0676.35032

[23] Y.Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations 23 no. 3–4 (1998), 487-545 | MR 1620632 | Zbl 0898.35004

[24] Y.Y. Li, L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equation, Comm. Pure Appl. Math. 51 (1998), 1445-1490 | MR 1639159 | Zbl 0933.35083

[25] C.S. Lin, W.M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations 72 (1988), 1-27 | MR 929196 | Zbl 0676.35030

[26] A. Malchiodi, Concentration of solutions for some singularly perturbed Neumann problems, Geometric Analysis and PDEs, Lecture Notes in Math. vol. 1977, Springer, Dordrecht (2009), 63-115 | MR 2500524 | Zbl 1184.35024

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

[28] C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces, Appl. Math. Sci. vol. 129, Springer-Verlag, New York (1998) | MR 1483320 | Zbl 0884.33001

[29] C. Müller, Spherical Harmonics, Lecture Notes in Math. vol. 17, Springer-Verlag, Berlin/Heidelberg/New York (1966) | MR 199449 | Zbl 0138.05101

[30] W.M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 no. 1 (1998), 9-18 | MR 1490535 | Zbl 0917.35047

[31] W.M. Ni, X.B. Pan, I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 no. 1 (1992), 1-20 | Zbl 0785.35041

[32] W.M. Ni, I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991), 819-851 | MR 1115095 | Zbl 0754.35042

[33] W.M. Ni, I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281 | MR 1219814 | Zbl 0796.35056

[34] W.M. Ni, I. Takagi, E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model. Recent topics in mathematics moving toward science and engineering, Japan J. Indust. Appl. Math. 18 no. 2 (2001), 259-272 | Zbl 1200.35172

[35] W.M. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 no. 7 (1995), 731-768 | MR 1342381 | Zbl 0838.35009

[36] J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc. 354 (2002), 3117-3154 | MR 1897394 | Zbl 0992.35031

[37] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162 | MR 454365 | Zbl 0356.35028

[38] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci. 237 (1952), 37-72 | MR 3363444

[39] Z.Q. Wang, On the existence of multiple, single-peaked solutions for a semilinear Neumann problem, Arch. Ration. Mech. Anal. 120 no. 4 (1992), 375-399 | MR 1185568 | Zbl 0784.35035

[40] J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations 134 no. 1 (1997), 104-133 | MR 1429093 | Zbl 0873.35007

[41] J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 no. 2 (1996), 315-333 | MR 1404386 | Zbl 0865.35011