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 , p. 219-259
Zbl 1150.35006 | MR 2244699
URL stable : http://www.numdam.org/item?id=ASNSP_2006_5_5_2_219_0

Classification:  35B40,  35B45,  35J55,  92C15,  92C40
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.

Bibliographie

[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 1943093 | Zbl 1010.47036

[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 1667283 | Zbl 1157.35407

[3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283-293. MR 1659454 | Zbl 0926.35125

[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 677997 | Zbl 0513.35007

[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 1132797 | Zbl 0753.35034

[6] G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations 94 (2004), 1-31. MR 2075433 | Zbl 1064.35180

[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 2048185 | Zbl 1183.35125

[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 1758878 | Zbl 0958.35054

[9] E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), 241-262. MR 1696122 | Zbl 0933.35070

[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 1757072 | Zbl 0948.35055

[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 2040640 | Zbl 1115.35039

[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 1911512 | Zbl 1109.35041

[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 2099569 | Zbl 1064.35182

[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 2269587 | Zbl 1165.35356

[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 2334995 | Zbl pre05181994

[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 2176935 | Zbl 1096.35017

[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 2183857 | Zbl 1207.35129

[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 1736974 | Zbl 0932.35080

[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 1742305 | Zbl 0942.35058

[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 688279 | Zbl 0506.35035

[21] H. Egnell, Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Differential Equations, 98 (1992), 34-56. MR 1168970 | Zbl 0778.35009

[22] L. C. Evans “Partial Differential Equations”, American Mathematical Society, Providence, Rhode Island, 1998. Zbl 1194.35001

[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 634248 | Zbl 0469.35052

[24] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer Verlag Berlin Heidelberg, 2001. MR 1814364 | Zbl 0361.35003

[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 1785679 | Zbl 0989.35054

[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 1782991 | Zbl 0964.35047

[27] C. Gui and J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differential Equations 158 (1999), 1-27. MR 1721719 | Zbl 1061.35502

[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 1758231 | Zbl 0949.35052

[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 1743431 | Zbl 0944.35020

[30] L. L. Helms, “Introduction to Potential Theory”, John Wiley & sons Inc., New York, 1969. MR 261018 | Zbl 0188.17203

[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 969899 | Zbl 0676.35032

[32] Y. Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Commun. Partial Differential Equations 23 (1998), 487-545. MR 1620632 | Zbl 0898.35004

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

[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 1219814 | Zbl 0796.35056

[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 1115095 | Zbl 0754.35042

[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 1639546 | Zbl 0946.35007

[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 1342381 | Zbl 0838.35009

[38] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005), 141-164. MR 2110455 | Zbl 1074.81023

[39] J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations 134 (1997), 104-133. MR 1429093 | Zbl 0873.35007

[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 1404386 | Zbl 0865.35011

[41] J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations, Manuscripta Math. 96 (1998), 113-131. MR 1624364 | Zbl 0901.35003

[42] X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Analysis TMA 12 (1998), 1297-1316. MR 969507 | Zbl 0671.35023