Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259.

We consider the problemwhere Ω 3 is a smooth and bounded domain, ε,γ 1 ,γ 2 >0, v,V:Ω, f:. We prove that this system has a least-energy solution v ε which develops, as ε0 + , a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches the most curved part of Ω, i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solutions in powers of ε up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in Ω.

Classification : 35B40, 35B45, 35J55, 92C15, 92C40
@article{ASNSP_2006_5_5_2_219_0,
     author = {D{\textquoteright}Aprile, Teresa},
     title = {Locating the boundary peaks of least-energy solutions to a singularly perturbed {Dirichlet} problem},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {219--259},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {2},
     year = {2006},
     mrnumber = {2244699},
     zbl = {1150.35006},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2006_5_5_2_219_0/}
}
TY  - JOUR
AU  - D’Aprile, Teresa
TI  - Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 219
EP  - 259
VL  - 5
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2006_5_5_2_219_0/
LA  - en
ID  - ASNSP_2006_5_5_2_219_0
ER  - 
%0 Journal Article
%A D’Aprile, Teresa
%T Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 219-259
%V 5
%N 2
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2006_5_5_2_219_0/
%G en
%F ASNSP_2006_5_5_2_219_0
D’Aprile, Teresa. Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259. http://archive.numdam.org/item/ASNSP_2006_5_5_2_219_0/

[1] P. W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal. 196 (2002), 211-264. | MR | Zbl

[2] P. W. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations 4 (1999), 1-69. | MR | Zbl

[3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283-293. | MR | Zbl

[4] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549-561. | MR | Zbl

[5] C. C. Chen and C. S. Lin, Uniqueness of the ground state solution of Δu+f(u)=0 in N ,N3, Comm. Partial Differential Equations 16 (1991), 1549-1572. | MR | Zbl

[6] G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations 94 (2004), 1-31. | MR | Zbl

[7] E. N. Dancer, Stable and finite Morse index solutions on N or on bounded domains with small diffusion. II, Indiana Univ. Math. J. 53 (2004), 97-108. | MR | Zbl

[8] E. N. Dancer and S. Yan, Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem, Topol. Methods Nonlinear Anal. 14 (1999), 1-38. | MR | Zbl

[9] E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), 241-262. | MR | Zbl

[10] E. N. Dancer and S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), 1177-1212. | MR | Zbl

[11] E. N. Dancer and S. Yan, Peak solutions for an elliptic system of Fitzhugh-Nagumo type, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003), 679-709. | Numdam | MR | Zbl

[12] E. N. Dancer and S. Yan, On the profile of the changing-sign mountain pass solutions for an elliptic problem, Trans. Amer. Math. Soc. 354 (2002), 3573-3600. | MR | Zbl

[13] T. D'Aprile and D. Mugnai, Existence of solitary waves for the nonlinear Klein-Gordon Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893-906. | MR | Zbl

[14] T. D'Aprile and J. Wei, Boundary concentration in radial solutions for a system of semilinear elliptic equations, J. London Math. Soc. (2), to appear. | MR | Zbl

[15] T. D'Aprile and J. Wei, Clustered solutions around harmonic centers to a coupled elliptic system, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. | Numdam | MR

[16] T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal. 37 (2005), 321-342. | MR | Zbl

[17] T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations 25 (2006), 105-137. | MR | Zbl

[18] M. Del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), 883-898. | MR | Zbl

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

[20] M. J. Esteban and P. L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 1-14. | MR | Zbl

[21] H. Egnell, Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Differential Equations, 98 (1992), 34-56. | MR | Zbl

[22] L. C. Evans “Partial Differential Equations”, American Mathematical Society, Providence, Rhode Island, 1998. | Zbl

[23] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in N , In: “Mathematical Analysis and Applications”, Part A, N. Nachbin (ed.), Adv. Math. Suppl. Stud., Vol. 7, 1981, 369-402. | MR | Zbl

[24] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer Verlag Berlin Heidelberg, 2001. | MR | Zbl

[25] M. Grossi and A. Pistoia, On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differential Equations 5 (2000), 1397-1420. | MR | Zbl

[26] M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2000), 143-175. | MR | Zbl

[27] C. Gui and J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differential Equations 158 (1999), 1-27. | MR | Zbl

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

[29] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 249-289. | Numdam | MR | Zbl

[30] L. L. Helms, “Introduction to Potential Theory”, John Wiley & sons Inc., New York, 1969. | MR | Zbl

[31] M. K. Kwong, Uniqueness of positive solutions of Δu-u+u p =0 in N , Arch. Ration. Mech. Anal. 105 (1991), 243-266. | MR | Zbl

[32] Y. Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Commun. Partial Differential Equations 23 (1998), 487-545. | MR | Zbl

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

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

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

[36] W. M. Ni, I. Takagi and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions, Duke Math. J. 94 (1998), 597-618. | MR | Zbl

[37] W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 723-761. | MR | Zbl

[38] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005), 141-164. | MR | Zbl

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

[40] J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), 315-333. | MR | Zbl

[41] J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations, Manuscripta Math. 96 (1998), 113-131. | MR | Zbl

[42] X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Analysis TMA 12 (1998), 1297-1316. | MR | Zbl