Concentration on curves for nonlinear Schrödinger equations. (English) Zbl 1123.35003

Del Pino, Manuel; Kowalczyk, Michal; Wei, Jun-Cheng

Commun. Pure Appl. Math. 60, No. 1, 113-146 (2007).

Suppose that \(p>1\) and \(V\colon\mathbb R^N\to\mathbb R\) is smooth and such that \(\inf_{x\in\mathbb R^N}V(x)>0\). In the well known article [A. Ambrosetti, A. Malchiodi, and W.-M. Ni, Commun. Math. Phys. 235, No. 3, 427–466 (2003; Zbl 1072.35019)] it is proved that the problem \[ -\varepsilon^2 \Delta u + V(x) u=u^p\tag{1} \]
possesses positive spike layer solutions concentrating near spheres of radius \(r_0\) as \(\varepsilon\to0\) if \(V\) is radially symmetric and if the function \(r\mapsto r^{N-1}V^\sigma(r)\) has a strict local extremum at \(r_0\), with \(\sigma:=\frac{p+1}{p-1}-\frac12\). Without imposing the condition of radial symmetry on \(V\), the present work generalizes that result in the case \(N=2\) (and proves a related conjecture exposed in that article) to the existence of positive solutions of (1) concentrating near nondegenerate closed geodesics of the weighted metric \(V^\sigma(dx_1^2+dx_2^2)\) in \(\mathbb R^2\). Here it has to be assumed that \(\varepsilon\) is small enough and satisfies a gap condition
\[ | \varepsilon^2k^2-\lambda_*| \geq c\varepsilon,\qquad\forall k\in\mathbb N, \]
with some constants \(c,\lambda_*>0\). This condition implies bounds for the inverse of a linear differential operator that arises in the finite dimensional reduction.

Reviewer: Nils Ackermann (México, D.F.)

MSC:

35B25	Singular perturbations in context of PDEs
35J60	Nonlinear elliptic equations
35Q55	NLS equations (nonlinear Schrödinger equations)
47F05	General theory of partial differential operators

Keywords:

spike layer solution; semiclassical limit; standing wave

Citations:

Zbl 1072.35019

References:

[1]	Alikakos, Calc Var Partial Differential Equations 11 pp 233– (2000)
[2]	Ambrosetti, Arch Rational Mech Anal 140 pp 285– (1997)
[3]	Ambrosetti, Comm Math Phys 235 pp 427– (2003)
[4]	Ambrosetti, Arch Ration Mech Anal 159 pp 253– (2001)
[5]	Badiale, Nonlinear Anal 49 pp 947– (2002)
[6]	Benci, J Differential Equations 184 pp 109– (2002)
[7]	Byeon, Arch Ration Mech Anal 165 pp 295– (2002)
[8]	Cingolani, J Differential Equations 160 pp 118– (2000)
[9]	del Pino, Calc Var Partial Differential Equations 4 pp 121– (1996)
[10]	del Pino, J Funct Anal 149 pp 245– (1997)
[11]	del Pino, Ann Inst H Poincaré Anal Non Linéaire 15 pp 127– (1998)
[12]	del Pino, Math Ann 324 pp 1– (2002)
[13]	Felmer, Commun Contemp Math 4 pp 481– (2002)
[14]	Floer, J Funct Anal 69 pp 397– (1986)
[15]	Grossi, Ann Inst H Poincaré Anal Non Linéaire 19 pp 261– (2002)
[16]	Gui, Comm Partial Differential Equations 21 pp 787– (1996)
[17]	Jeanjean, Calc Var Partial Differential Equations 21 pp 287– (2004) · Zbl 1060.35012 · doi:10.1007/s00526-003-0261-6
[18]	Kang, Adv Differential Equations 5 pp 899– (2000)
[19]	Approximate invariant manifold of the Allen-Cahn flow in two dimensions. Partial differential equations and inverse problems, 233–239. Contemporary Mathematics, 362. American Mathematical Society, Providence, R.I., 2004. · doi:10.1090/conm/362/06616
[20]	Kowalczyk, Ann Mat Pura Appl (4) 184 pp 17– (2005)
[21]	; Sturm-Liouville and Dirac operators. Mathematics and Its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. · doi:10.1007/978-94-011-3748-5
[22]	Li, Adv Differential Equations 2 pp 955– (1997)
[23]	Lin, J Differential Equations 72 pp 1– (1988)
[24]	Malchiodi, C R Math Acad Sci Paris 338 pp 775– (2004) · Zbl 1081.35044 · doi:10.1016/j.crma.2004.03.023
[25]	Malchiodi, Geom Funct Anal 15 pp 1162– (2005)
[26]	Malchiodi, Comm Pure Appl Math 55 pp 1507– (2002)
[27]	Malchiodi, Duke Math J 124 pp 105– (2004)
[28]	; Foliations by mean curvature sets. Preprint, 2005.
[29]	Ni, Comm Pure Appl Math 44 pp 819– (1991)
[30]	Ni, Duke Math J 70 pp 247– (1993)
[31]	Oh, Comm Math Phys 121 pp 11– (1989)
[32]	Oh, Comm Math Phys 131 pp 223– (1990)
[33]	Pacard, J Differential Geom 64 pp 359– (2003)
[34]	Rabinowitz, Z Angew Math Phys 43 pp 270– (1992)
[35]	Shatah, J Differential Equations 186 pp 572– (2002)
[36]	Sirakov, Ann Mat Pura Appl (4) 181 pp 73– (2002)
[37]	Wang, Comm Math Phys 153 pp 229– (1993)
[38]	Wang, J Differential Equations 159 pp 102– (1999)