Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps

Cerami, Giovanna; Passaseo, Donato; Solimini, Sergio

Cerami, Giovanna ; Passaseo, Donato ; Solimini, Sergio

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 23-40 / Harvested from Numdam

Résumé
Abstract

Dans ce papier nous considérons l'équation $(E) - Δ u + a (x) u = {| u |}^{p - 1} u en ℝ^{N},$ où $N ⩾ 2$ , $p > 1$ , $p < 2^{⁎} - 1 = \frac{N + 2}{N - 2}$ , si $N ⩾ 3$ . Pendant les trente dernières années la question de l'existence et de la multiplicité de solutions d'(E) a été largement examinée surtout conformément aux suppositions de symétrie sur a. Le but de ce papier est de montrer que, différemment de ceux trouvés conformément à la supposition de symétrie, les solutions trouvées dans [6] admettent une configuration de limite et donc (E) admet aussi une solution positive d'énergie infinie ayant une infinité de ‘bumps’.

In this paper we consider the equation $(E) - Δ u + a (x) u = {| u |}^{p - 1} u in ℝ^{N},$ where $N ⩾ 2$ , $p > 1$ , $p < 2^{⁎} - 1 = \frac{N + 2}{N - 2}$ , if $N ⩾ 3$ . During last thirty years the question of the existence and multiplicity of solutions to (E) has been widely investigated mostly under symmetry assumptions on a. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in [6] admit a limit configuration and so (E) also admits a positive solution of infinite energy having infinitely many ‘bumps’.

Publié le : 2015-01-01

MR 3303940 | Zbl 1311.35081 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2013.08.008
Classification: 35J20, 35J60, 35Q55

@article{AIHPC_2015__32_1_23_0,
     author = {Cerami, Giovanna and Passaseo, Donato and Solimini, Sergio},
     title = {Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {23-40},
     doi = {10.1016/j.anihpc.2013.08.008},
     mrnumber = {3303940},
     zbl = {1311.35081},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_1_23_0}
}

Cerami, Giovanna; Passaseo, Donato; Solimini, Sergio. Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 23-40. doi : 10.1016/j.anihpc.2013.08.008. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_1_23_0/

Bibliographie

[1] A. Bahri, Y.Y. Li, On a min–max procedure for the existence of a positive solution for certain scalar field equations in $R^{N}$ , Rev. Mat. Iberoam. 6 (1990), 1 -15 | MR 1086148 | Zbl 0731.35036

[2] A. Bahri, P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 (1997), 365 -413 | Numdam | MR 1450954 | Zbl 0883.35045

[3] H. Brezis, Analyse fonctionelle. Theorie et applications, Collect. Math. Appl. Maîtrise , Masson, Paris (1983) | MR 697382

[4] G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math. 74 (2006), 47 -77 | MR 2278729 | Zbl 1121.35054

[5] G. Cerami, G. Devillanova, S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differ. Equ. 23 (2005), 139 -168 | MR 2138080 | Zbl 1078.35113

[6] G. Cerami, D. Passaseo, S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Commun. Pure Appl. Math. 66 (2013), 372 -413 | MR 3008228 | Zbl 1292.35128

[7] P.L. Lions, The concentration compactness principle in the calculus of variations, Parts I and II, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 109 -145 | Numdam | Numdam | MR 778970 | Zbl 0541.49009

[8] J. Wei, S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $ℝ^{N}$ , Calc. Var. Partial Differ. Equ. 37 (2010), 423 -439 | MR 2592980 | Zbl 1189.35106