Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

D’Aprile, Teresa

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259

We consider the problemwhere $\Omega \subset {ℝ}^3$ is a smooth and bounded domain, $\epsilon ,{\gamma }_1,{\gamma }_2>0,$ $v,V:\Omega \to ℝ$, $f:ℝ\to ℝ$. We prove that this system has a least-energy solution $v_{\epsilon }$ which develops, as $\epsilon \to 0^{+}$, 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 $\partial \Omega$, 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 $\epsilon$ up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in $\Omega$.

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

[5] C. C. Chen and C. S. Lin, Uniqueness of the ground state solution of $\Delta u+f\left(u\right)=0$ in ${ℝ}^N,N\ge 3$, Comm. Partial Differential Equations 16 (1991), 1549-1572. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | MR | Numdam

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

[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 ${ℝ}^N$, In: “Mathematical Analysis and Applications”, Part A, N. Nachbin (ed.), Adv. Math. Suppl. Stud., Vol. 7, 1981, 369-402. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

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