We study the existence of positive solutions on to semilinear elliptic equation where and f is modeled on the power case . Denoting with c the mountain pass level of , (), we show, via a new energy constrained variational argument, that for any there exists a positive bounded solution such that and as uniformly with respect to . We also characterize the monotonicity, symmetry and periodicity properties of .
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}, pages = {725--749}, publisher = {Elsevier}, volume = {31}, number = {4}, year = {2014}, doi = {10.1016/j.anihpc.2013.07.003}, mrnumber = {3249811}, zbl = {06349267}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.003/} }
TY - JOUR AU - Alessio, Francesca AU - Montecchiari, Piero TI - An energy constrained method for the existence of layered type solutions of NLS equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 725 EP - 749 VL - 31 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.003/ DO - 10.1016/j.anihpc.2013.07.003 LA - en ID - AIHPC_2014__31_4_725_0 ER -
%0 Journal Article %A Alessio, Francesca %A Montecchiari, Piero %T An energy constrained method for the existence of layered type solutions of NLS equations %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 725-749 %V 31 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.003/ %R 10.1016/j.anihpc.2013.07.003 %G en %F 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/articles/10.1016/j.anihpc.2013.07.003/
[1] Stationary layered solutions for a system of Allen–Cahn type equations, Indiana Univ. Math. J. (2013) | MR | Zbl
,[2] Stationary layered solutions in for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 no. 2 (2000), 177 -202 | MR | Zbl
, , ,[3] Entire solutions in for a class of Allen–Cahn equations, ESAIM Control Optim. Calc. Var. 11 (2005), 633 -672 | EuDML | Numdam | MR | Zbl
, ,[4] Multiplicity of entire solutions for a class of almost periodic Allen–Cahn type equations, Adv. Nonlinear Stud. 5 (2005), 515 -549 | MR | Zbl
, ,[5] Brake orbits type solutions to some class of semilinear elliptic equations, Calc. Var. Partial Differential Equations 30 no. 51 (2007), 83 | MR | Zbl
, ,[6] A Primer of Nonlinear Analysis, Cambridge University Press (1995) | MR | Zbl
, ,[7] 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 | Zbl
, ,[8] Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313 -345 | MR | Zbl
, ,[9] 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 | Zbl
, , ,[10] Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. 13 (2001), 181 -211 | MR | Zbl
, ,[11] An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Mathematicos vol. 26 , IM-UFRJ, Rio de Janeiro (1989)
,[12] Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 no. 4 (1982), 549 -561 | MR | Zbl
, ,[13] New solutions of equations on , Ann. Sc. Norm. Super. Pisa Cl. Sci. 30 (2001), 535 -563 | EuDML | Numdam | MR | Zbl
,[14] The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math. 224 no. 4–10 (2010), 1462 -1516 | MR | Zbl
, , ,[15] Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 no. 4 (2008), 904 -933 | MR | Zbl
,[16] Axial symmetry of some steady state solutions to nonlinear Schroedinger equations, Proc. Amer. Math. Soc. 139 no. 3 (2011), 1023 -1032 | MR | Zbl
, , ,[17] A remark on least energy solutions in , Proc. Amer. Math. Soc. 131 (2003), 2399 -2408 | MR | Zbl
, ,[18] Uniqueness of positive solutions of in , Arch. Ration. Mech. Anal. 105 no. 3 (1989), 243 -266 | MR | Zbl
,[19] Radial symmetry of positive solutions of nonlinear elliptic equations in , Comm. Partial Differential Equations 18 no. 5–6 (1993), 1043 -1054 | MR | Zbl
, ,[20] Symétrie et Compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315 -334 | MR | Zbl
,[21] Some new entire solutions of semilinear elliptic equations on , Adv. Math. 221 no. 6 (2009), 1843 -1909 | MR | Zbl
,[22] Multiplicity results for a class of semilinear elliptic equations on , Rend. Semin. Mat. Univ. Padova 95 (1996), 1 -36 | EuDML | Numdam | MR
,[23] Analysis, Grad. Stud. Math. vol. 14 , American Mathematical Society, Providence, RI (2001) | MR
, ,[24] Lectures on Linear Partial Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 17 , American Mathematical Society (1973) | MR | Zbl
,[25] On Elliptic partial differential equations, Ann. Sc. Norm. Pisa 13 (1959), 116 -162 | EuDML | Numdam | MR | Zbl
,[26] 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
,[27] Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 no. 3 (2000), 897 -923 | MR | Zbl
, ,[28] The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Appl. Math. Sci. vol. 139 , Springer-Verlag, New York (1999) | MR | Zbl
, ,[29] Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 -162 | MR | Zbl
,[30] Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 no. 4 (1983), 567 -576 | MR | Zbl
,Cited by Sources: