Concentrating standing waves for Davey-Stewartson systems. (English) Zbl 1505.35181
Summary: In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey-Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey-Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey-Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Ambrosetti, A., Badiale, M. and Cingolani, S.. Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140 (1997), 285-300. · Zbl 0896.35042 |
| [2] | Berestycki, H., Gallouët, T. and Kavian, O.. Équations de champs scalaires Euclidens non linéaires dans le plan. C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307-310. · Zbl 0544.35042 |
| [3] | Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations, I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 (1983), 313-345. · Zbl 0533.35029 |
| [4] | Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations, II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82 (1983), 347-375. · Zbl 0556.35046 |
| [5] | Brézis, H. and Lieb, E. H.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 486-490. · Zbl 0526.46037 |
| [6] | Byeon, J. and Jeanjean, L.. Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185 (2007), 185-200. · Zbl 1132.35078 |
| [7] | Byeon, J. and Jeanjean, L.. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Contin. Dyn. Syst. Ser. A19 (2007), 255-269. · Zbl 1155.35089 |
| [8] | Byeon, J., Jeanjean, L. and Tanaka, K.. Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Commun. Partial Differ. Equ. 33 (2008), 1113-1136. · Zbl 1155.35344 |
| [9] | Byeon, J. and Tanaka, K.. Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15 (2013), 1859-1899. · Zbl 1294.35131 |
| [10] | Byeon, J. and Tanaka, K.. Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations. Mem. Amer. Math. Soc. 229 (2014), No. 1076. · Zbl 1303.35094 |
| [11] | Byeon, J. and Wang, Z.-Q.. Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18 (2003), 207-219. · Zbl 1073.35199 |
| [12] | Carles, R., Dumas, E. and Sparber, C.. Geometric optics and instability for NLS and Davey-Stewartson models. J. Eur. Math. Soc. 14 (2012), 1885-1921. · Zbl 1273.35248 |
| [13] | Cingolani, S., Jeanjean, L. and Tanaka, K.. Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well. Calc. Var. Partial Differ. Equ. 53 (2015), 413-439. · Zbl 1326.35333 |
| [14] | Cingolani, S. and Lazzo, N.. Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10 (1997), 1-13. · Zbl 0903.35018 |
| [15] | Cipolatti, R.. On the existence of standing waves for a Davey-Stewartson system. Commun. Partial Differ. Equ. 17 (1992), 967-988. · Zbl 0762.35109 |
| [16] | Cipolatti, R.. On the instability of ground states for a Davey-Stewartson system. Ann. Inst. Henri Poincaré Phys. Théor. 58 (1993), 85-104. · Zbl 0784.35106 |
| [17] | Coti Zelati, V. and Rabinowitz, P.. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693-727. · Zbl 0744.34045 |
| [18] | Craig, W., Schanz, U. and Sulem, C.. The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1999), 131-175. · Zbl 0892.76008 |
| [19] | D’Aprile, T. and Ruiz, D.. Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems. Math. Z. 268 (2011), 605-634. · Zbl 1228.35019 |
| [20] | D’Avenia, P., Pomponio, A. and Ruiz, D.. Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods. J. Funct. Anal. 262 (2012), 4600-4633. · Zbl 1239.35152 |
| [21] | Davey, A. and Stewartson, K.. On three-dimensional packets of surface waves. Proc. Roy. Soc. Lond. Ser. A338 (1974), 101-110. · Zbl 0282.76008 |
| [22] | Del Pino, M. and Felmer, P. L.. Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996), 121-137. · Zbl 0844.35032 |
| [23] | Del Pino, M. and Felmer, P. L.. Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non. Linéaire15 (1998), 127-149. · Zbl 0901.35023 |
| [24] | Figueiredo, G. M., Ikoma, N. and Santos, J. R. Jr. Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213 (2014), 931-979. · Zbl 1302.35356 |
| [25] | Floer, A. and Weinstein, A.. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), 397-408. · Zbl 0613.35076 |
| [26] | Gan, Z. H. and Zhang, J.. Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system. Commun. Math. Phys. 283 (2008), 93-125. · Zbl 1158.35416 |
| [27] | Ghidaglia, J. M. and Saut, J. C.. On the initial value problem for the Davey-Stewartson systems. Nonlinearity3 (1990), 475-506. · Zbl 0727.35111 |
| [28] | Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn, , vol. 224 (Berlin: Springer, 1983). · Zbl 0562.35001 |
| [29] | Gui, C.. Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21 (1996), 787-820. · Zbl 0857.35116 |
| [30] | Guo, B. L. and Wang, B. X.. The Cauchy problem for Davey-Stewartson systems. Commun Pure Appl. Math. 52 (1999), 1477-1490. · Zbl 1026.35096 |
| [31] | Jeanjean, L. and Tanaka, K.. A remark on least energy solutions in \({\mathbb{R}}^N \). Proc. Amer. Math. Soc. 131 (2002), 2399-2408. · Zbl 1094.35049 |
| [32] | Lieb, E. H. and Loss, M.. Analysis, 2nd edn, , vol. 14 (Providence, RI: American Mathematical Society, 2001). · Zbl 0966.26002 |
| [33] | Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. H. Poincaré Anal. Non. Linéaire2 (1984), 223-283. · Zbl 0704.49004 |
| [34] | Oh, Y. G.. Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class \(( V )_a\). Commun. Partial Differ. Equ. 13 (1988), 1499-1519. · Zbl 0702.35228 |
| [35] | Oh, Y. G.. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131 (1990), 223-253. · Zbl 0753.35097 |
| [36] | Ohta, M.. Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in \(\mathbb{R}^2 \). Ann. Inst. Henri Poincaré Phys. Théor. 63 (1995), 111-117. · Zbl 0832.35132 |
| [37] | Ohta, M.. Instability of standing waves for the generalized Davey-Stewartson system. Ann. Inst. Henri Poincaré Phys. Théor. 62 (1995), 69-80. · Zbl 0839.35012 |
| [38] | Ozawa, T.. Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. Proc. Roy. Soc. Lond. Ser. A436 (1992), 345-349. · Zbl 0754.35114 |
| [39] | Pucci, P. and Serrin, J.. A general variational identity. Indiana Univ. Math. J. 35 (1986), 681-703. · Zbl 0625.35027 |
| [40] | Rabinowitz, P.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270-291. · Zbl 0763.35087 |
| [41] | Wang, X.. On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153 (1993), 229-244. · Zbl 0795.35118 |
| [42] | Weinstein, M.. Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87 (1983), 567-576. · Zbl 0527.35023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
