An energy constrained method for the existence of layered type solutions of NLS equations
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, p. 725-749
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We study the existence of positive solutions on N+1 to semilinear elliptic equation -Δu+u=f(u) where N1 and f is modeled on the power case f(u)=|u| p-1 u. Denoting with c the mountain pass level of V(u)=1 2u H 1 ( N ) 2 - N F(u)dx, uH 1 ( N ) (F(s)= 0 s f(t)dt), we show, via a new energy constrained variational argument, that for any b[0,c) there exists a positive bounded solution v b C 2 ( N+1 ) such that E v b (y)=1 2 y v b (·,y) L 2 ( N ) 2 -V(v b (·,y))=-b and v(x,y)0 as |x|+ uniformly with respect to y. We also characterize the monotonicity, symmetry and periodicity properties of v b .

DOI : https://doi.org/10.1016/j.anihpc.2013.07.003
Classification:  35J60,  35B08,  35B40,  35J20,  34C37
Keywords: Semilinear elliptic equations, Locally compact case, Variational methods, Energy constraints
@article{AIHPC_2014__31_4_725_0,
     author = {Alessio, Francesca and Montecchiari, Piero},
     title = {An energy constrained method for the existence of layered type solutions of NLS equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {4},
     year = {2014},
     pages = {725-749},
     doi = {10.1016/j.anihpc.2013.07.003},
     zbl = {06349267},
     mrnumber = {3249811},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_4_725_0}
}
Alessio, Francesca; Montecchiari, Piero. An energy constrained method for the existence of layered type solutions of NLS equations. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 725-749. doi : 10.1016/j.anihpc.2013.07.003. http://www.numdam.org/item/AIHPC_2014__31_4_725_0/

[1] F. Alessio, Stationary layered solutions for a system of Allen–Cahn type equations, Indiana Univ. Math. J. (2013) | MR 3188554 | Zbl 1300.35035

[2] F. Alessio, L. Jeanjean, P. Montecchiari, Stationary layered solutions in 2 for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 no. 2 (2000), 177 -202 | MR 1782992 | Zbl 0965.35050

[3] F. Alessio, P. Montecchiari, Entire solutions in 2 for a class of Allen–Cahn equations, ESAIM Control Optim. Calc. Var. 11 (2005), 633 -672 | Numdam | MR 2167878 | Zbl 1084.35020

[4] F. Alessio, P. Montecchiari, Multiplicity of entire solutions for a class of almost periodic Allen–Cahn type equations, Adv. Nonlinear Stud. 5 (2005), 515 -549 | MR 2180581 | Zbl 1284.35170

[5] F. Alessio, P. Montecchiari, Brake orbits type solutions to some class of semilinear elliptic equations, Calc. Var. Partial Differential Equations 30 no. 51 (2007), 83 | MR 2333096 | Zbl 1153.35025

[6] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press (1995) | MR 1336591 | Zbl 0818.47059

[7] H. Berestycki, T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires, C. R. Acad. Sci. Paris 293 no. 9 (1981), 489 -492 | MR 646873 | Zbl 0492.35010

[8] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313 -345 | MR 695535 | Zbl 0533.35029

[9] H. Berestycki, T. Gallouët, O. Kavian, Équations de Champs scalaires Euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math. 297 no. 5 (1983), 307 -310 | MR 734575 | Zbl 0544.35042

[10] J. Busca, P. Felmer, Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. 13 (2001), 181 -211 | MR 1861097 | Zbl 1151.35344

[11] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Mathematicos vol. 26 , IM-UFRJ, Rio de Janeiro (1989)

[12] T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 no. 4 (1982), 549 -561 | MR 677997 | Zbl 0513.35007

[13] E.N. Dancer, New solutions of equations on n , Ann. Sc. Norm. Super. Pisa Cl. Sci. 30 (2001), 535 -563 | Numdam | MR 1896077 | Zbl 1025.35009

[14] M. Del Pino, M. Kowalczyk F Pacard, J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math. 224 no. 4–10 (2010), 1462 -1516 | MR 2646302 | Zbl 1197.35114

[15] C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 no. 4 (2008), 904 -933 | MR 2381198 | Zbl 1148.35023

[16] C. Gui, A. Malchiodi, H. Xu, Axial symmetry of some steady state solutions to nonlinear Schroedinger equations, Proc. Amer. Math. Soc. 139 no. 3 (2011), 1023 -1032 | MR 2745653 | Zbl 1211.35115

[17] L. Jeanjean, K. Tanaka, A remark on least energy solutions in N , Proc. Amer. Math. Soc. 131 (2003), 2399 -2408 | MR 1974637 | Zbl 1094.35049

[18] M.K. Kwong, Uniqueness of positive solutions of Δu-u+u p in n , Arch. Ration. Mech. Anal. 105 no. 3 (1989), 243 -266 | MR 969899 | Zbl 0676.35032

[19] Y. Li, W. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in N , Comm. Partial Differential Equations 18 no. 5–6 (1993), 1043 -1054 | MR 1218528 | Zbl 0788.35042

[20] P.L. Lions, Symétrie et Compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315 -334 | MR 683027 | Zbl 0501.46032

[21] A. Malchiodi, Some new entire solutions of semilinear elliptic equations on n , Adv. Math. 221 no. 6 (2009), 1843 -1909 | MR 2522830 | Zbl 1178.35186

[22] P. Montecchiari, Multiplicity results for a class of semilinear elliptic equations on m , Rend. Semin. Mat. Univ. Padova 95 (1996), 1 -36 | Numdam | MR 1405365

[23] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math. vol. 14 , American Mathematical Society, Providence, RI (2001) | MR 1817225

[24] L. Nirenberg, Lectures on Linear Partial Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 17 , American Mathematical Society (1973) | MR 450755 | Zbl 0267.35001

[25] L. Nirenberg, On Elliptic partial differential equations, Ann. Sc. Norm. Pisa 13 (1959), 116 -162 | Numdam | MR 109940 | Zbl 0088.07601

[26] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 65 , American Mathematical Society, Providence, RI (1986) | MR 845785

[27] J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 no. 3 (2000), 897 -923 | MR 1803216 | Zbl 0979.35049

[28] C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Appl. Math. Sci. vol. 139 , Springer-Verlag, New York (1999) | MR 1696311 | Zbl 0867.35094

[29] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 -162 | MR 454365 | Zbl 0356.35028

[30] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 no. 4 (1983), 567 -576 | MR 691044 | Zbl 0527.35023