Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary. (English) Zbl 1390.35319

Clapp, Mónica; Fernández, Yéferson

J. Fixed Point Theory Appl. 20, No. 1, Paper No. 48, 26 p. (2018).

Summary: We consider the problem \[ -\Delta u+(V_{\infty }+V(x)) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \] where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\geq 3\), \(V_{\infty}>0\), \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N})\), \(\inf _{\mathbb {R}^{N}}V>-V_{\infty}\) and \(2<p<\frac{2N}{N-2}\). We assume \(V\) is periodic in the first \(m\) variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega\) is periodic in the first \(m\) variables and has bounded complement in the other ones. Then, assuming that \(\Omega\) and \(V\) are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35J20	Variational methods for second-order elliptic equations
35B06	Symmetries, invariants, etc. in context of PDEs

Keywords:

nonlinear Schrödinger equation; partially periodic potential; exterior domain with unbounded boundary; sign-changing solutions; Lusternik-Schnirelmann theory for noncompact groups

References:

[1]	Ackermann, N.: Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential. J. Differ. Equ. 246(4), 1470-1499 (2009) · Zbl 1156.35024 · doi:10.1016/j.jde.2008.10.016
[2]	Ackermann, N., Weth, T.: Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting. Commun. Contemp. Math. 7(3), 269-298 (2005) · Zbl 1070.35083 · doi:10.1142/S0219199705001763
[3]	Alama, S., Li, Y.Y.: On “multibump” bound states for certain semilinear elliptic equations. Indiana Univ. Math. J. 41(4), 983-1026 (1992) · Zbl 0796.35043 · doi:10.1512/iumj.1992.41.41052
[4]	Angenent, S.: The shadowing lemma for elliptic PDE. In: Dynamics of Infinite-Dimensional Systems (Lisbon, 1986). NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 37, pp. 7-22. Springer, Berlin (1987) · Zbl 0533.35029
[5]	Antonyan, Sergey A.: Extensorial properties of orbit spaces of proper group actions. In: II Iberoamerican conference on topology and its applications (Morelia, 1997). Topology Appl., vol. 98, no. 1-3, pp. 35-46 (1999) · Zbl 0942.54011
[6]	Bahri, A., Li, Y.Y.: On a min-max procedure for the existence of a positive solution for certain scalar field equations in \[\mathbb{R}^N\] RN. Rev. Mat. Iberoam. 6, 1-15 (1990) · Zbl 0731.35036 · doi:10.4171/RMI/92
[7]	Bárcenas, N.: Mountain pass theorem with infinite discrete symmetry. Osaka J. Math. 53(2), 331-351 (2016) · Zbl 1375.58012
[8]	Bartsch, T.: Topological Methods for Variational Problems with Symmetries. Lecture Notes in Mathematics, vol. 1560. Springer, Berlin (1993) · Zbl 0789.58001
[9]	Benci, V., Cerami, G.: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Ration. Mech. Anal. 99(4), 283-300 (1987) · Zbl 0635.35036 · doi:10.1007/BF00282048
[10]	Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ratio. Mech. Anal. 82(4), 313-345 (1983) · Zbl 0533.35029
[11]	Carvalho, J.S., Maia, L.A., Miyagaki, O.H.: A note on existence of antisymmetric solutions for a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 62(1), 67-86 (2011) · Zbl 1228.35219 · doi:10.1007/s00033-010-0070-7
[12]	Cerami, G.: Some nonlinear elliptic problems in unbounded domains. Milan J. Math. 74, 47-77 (2006) · Zbl 1121.35054 · doi:10.1007/s00032-006-0059-z
[13]	Cerami, G., Clapp, M.: Sign changing solutions of semilinear elliptic problems in exterior domains. Calc. Var. Partial Differ. Equ. 30(3), 353-367 (2007) · Zbl 1174.35026 · doi:10.1007/s00526-007-0092-y
[14]	Cerami, G., Molle, R.: Positive solutions for some Schrödinger equations having partially periodic potentials. J. Math. Anal. Appl. 359(1), 15-27 (2009) · Zbl 1169.35331 · doi:10.1016/j.jmaa.2009.05.011
[15]	Cerami, G., Molle, R., Passaseo, D.: Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(1), 41-60 (2007) · Zbl 1123.35017 · doi:10.1016/j.anihpc.2005.09.007
[16]	Cerami, G., Passaseo, D., Solimini, S.: Nonlinear scalar field equations: existence of a positive solution with infinitely many bumps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 23-40 (2015) · Zbl 1311.35081 · doi:10.1016/j.anihpc.2013.08.008
[17]	Cingolani, S., Clapp, M., Secchi, S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63(2), 233-248 (2012) · Zbl 1247.35141 · doi:10.1007/s00033-011-0166-8
[18]	Clapp, M., Puppe, D.: Critical point theory with symmetries. J. Reine Angew. Math. 418, 1-29 (1991) · Zbl 0722.58011
[19]	Clapp, M., Salazar, D.: Multiple sign changing solutions of nonlinear elliptic problems in exterior domains. Adv. Nonlinear Stud. 12(3), 427-443 (2012) · Zbl 1263.35116 · doi:10.1515/ans-2012-0301
[20]	Clapp, M., Weth, T.: Multiple solutions of nonlinear scalar field equations. Commun. Partial Differ. Equ. 29, 1533-1554 (2004) · Zbl 1140.35401 · doi:10.1081/PDE-200037766
[21]	Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \[\mathbb{R}^nRn\]. Commun. Pure Appl. Math. 45(10), 1217-1269 (1992) · Zbl 0785.35029 · doi:10.1002/cpa.3160451002
[22]	tom Dieck, T.: Transformation Groups. de Gruyter Studies in Mathematics, vol. 8. Walter de Gruyter & Co, Berlin (1987) · Zbl 0611.57002
[23]	Dold, A.: Lectures on Algebraic Topology. Die Grundlehren der mathematischen Wissenschaften. Band 200. Springer, New York (1972) · Zbl 0234.55001
[24]	Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324-353 (1974) · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0
[25]	Esteban, M.J., Lions, P.-L.: Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. R. Soc. Edinb. Sect. A 93, 1-14 (1982) · Zbl 0506.35035 · doi:10.1017/S0308210500031607
[26]	Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2), 139-174 (1978) · Zbl 0403.57001 · doi:10.1007/BF01390270
[27]	Gidas, B.; Ni, W-M; Nirenberg, L.; Nachbin, L. (ed.), Symmetry of positive solutions of nonlinear elliptic equations in \[{\mathbb{R}}^N\] RN, No. 7a, 369-402 (1981), New York · Zbl 0469.35052
[28]	Husemoller, D.: Fibre Bundles, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1975) · Zbl 0307.55015
[29]	Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I and II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109-145 (1984). (223-283) · Zbl 0541.49009 · doi:10.1016/S0294-1449(16)30428-0
[30]	Molle, R.: Positive solutions for a nonlinear elliptic problem with strong lack of compactness. J. Lond. Math. Soc. 74(2), 441-452 (2006) · Zbl 1194.35155 · doi:10.1112/S002461070602309X
[31]	Palais, R.S.: On the existence of slices for actions of non-compact Lie groups. Ann. Math. 2(73), 295-323 (1961) · Zbl 0103.01802 · doi:10.2307/1970335
[32]	Rabinowitz, P.H.: A Note on a Semilinear Elliptic Equation on \[{\mathbb{R}}^nRn\]. Nonlinear Analysis. Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., pp. 307-317. Pisa (1991) · Zbl 0836.35045
[33]	Wei, J., Yan, S.: Infinitely many positive solutions for the nonlinear Schrödinger equations in \[\mathbb{R}^N\] RN. Calc. Var. Partial Differ. Equ. 37(3-4), 423-439 (2010) · Zbl 1189.35106 · doi:10.1007/s00526-009-0270-1
[34]	Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston, Inc., Boston (1996) · Zbl 0856.49001