On the existence of infinitely many solutions for a class of semilinear elliptic equations in $ \mathbb{R}^{N} $

Alessio, Francesca; Caldiroli, Paolo; Montecchiari, Piero

On the existence of infinitely many solutions for a class of semilinear elliptic equations in

R^{N}

Alessio, Francesca ; Caldiroli, Paolo ; Montecchiari, Piero

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998), p. 157-165 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

We show, by variational methods, that there exists a set $A$ open and dense in $a \in L^{\infty} (R^{N}) : a \geq 0$ such that if $a \in A$ then the problem $- △ u + u = a (x) {|u|}^{p - 1} u, u \in H^{1} (R^{N})$ , with $p$ subcritical (or more general nonlinearities), admits infinitely many solutions.

Usando metodi variazionali, si dimostra che esiste un insieme $A$ aperto e denso in $a \in L^{\infty} (R^{N}) : a \geq 0$ tale che per ogni $a \in A$ il problema $- △ u + u = a (x) {|u|}^{p - 1} u, u \in H^{1} (R^{N})$ , con $p$ sottocritico (o con nonlinearità più generali), ammette infinite soluzioni.

Publié le : 1998-09-01

MR 1683006 | Zbl 0923.35057

@article{RLIN_1998_9_9_3_157_0,
     author = {Francesca Alessio and Paolo Caldiroli and Piero Montecchiari},
     title = {On the existence of infinitely many solutions for a class of semilinear elliptic equations in \( \mathbb{R}^{N} \)},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {9},
     year = {1998},
     pages = {157-165},
     zbl = {0923.35057},
     mrnumber = {1683006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1998_9_9_3_157_0}
}

Alessio, Francesca; Caldiroli, Paolo; Montecchiari, Piero. On the existence of infinitely many solutions for a class of semilinear elliptic equations in \( \mathbb{R}^{N} \). Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998) pp. 157-165. http://gdmltest.u-ga.fr/item/RLIN_1998_9_9_3_157_0/

Bibliographie

[1] Alama, S. - Li, Y. Y., Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Diff. Eq., 96, 1992, 88-115. | MR 1153310 | Zbl 0766.35009

[2] Alessio, F. - Caldiroli, P. - Montecchiari, P., Genericity of the existence of infinitely many solutions for a class of semilinear elliptic equations in $R^{n}$ . Ann. Scuola Norm. Sup. Pisa, Cl. Sci., (4), to appear. | MR 1831991 | Zbl 0931.35047

[3] Ambrosetti, A. - Badiale, M., Homoclinics: Poincaré-Melnikov type results via a variational approach. C.R. Acad. Sci. Paris, 323, s. I, 1996, 753-758; Ann. Inst. H. Poincaré, Anal. non linéaire, to appear. | MR 1416171 | Zbl 0887.34042

[4] Ambrosetti, A. - Badiale, M. - Cingolani, S., Semiclassical states of nonlinear Schrödinger equation. Arch. Rat. Mech. Anal., 140, 1997, 285-300. | MR 1486895 | Zbl 0896.35042

[5] Angenent, S., The Shadowing Lemma for Elliptic PDE. In: S. N. Chow - J. K. Hale (eds.), Dynamics of Infinite Dimensional Systems. NATO ASI Series, F37, Springer-Verlag, 1987. | MR 921893 | Zbl 0653.35030

[6] Bahri, A. - Li, Y. Y., On a Min-Max Procedure for the Existence of a Positive Solution for Certain Scalar Field Equation in $R^{n}$ . Rev. Mat. Iberoamericana, 6, 1990, 1-15. | MR 1086148 | Zbl 0731.35036

[7] Bahri, A. - Lions, P. L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré, Anal. non linéaire, 14, 1997, 365-413. | MR 1450954 | Zbl 0883.35045

[8] Berestycki, H. - Lions, P. L., Nonlinear scalar field equations. Arch. Rat. Mech. Anal., 82, 1983, 313-345. | MR 695535 | Zbl 0533.35029

[9] Cao, D. M., Positive solutions and bifurcation from the essential spectrum of a semilinear elliptic equation in $R^{n}$ . Nonlinear Anal. T.M.A., 15, 1990, 1045-1052. | MR 1082280 | Zbl 0729.35049

[10] Cao, D. M., Multiple solutions of a semilinear elliptic equation in $R^{n}$ . Ann. Inst. H. Poincaré, Anal. non linéaire, 10, 1993, 593-604. | MR 1253603 | Zbl 0797.35039

[11] Cao, D. M. - Noussair, E. S., Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $R^{n}$ . Ann. Inst. H. Poincaré, Anal. non linéaire, 13, 1996, 567-588. | MR 1409663 | Zbl 0859.35032

[12] Cingolani, S., On a perturbed semilinear elliptic equation in $R^{n}$ . Comm. Appl. Anal., to appear. | MR 1669769 | Zbl 0922.35051

[13] Coti Zelati, V. - Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on $R^{n}$ . Comm. Pure Appl. Math., 45, 1992, 1217-1269. | MR 1181725 | Zbl 0785.35029

[14] Del Pino, M. - Felmer, P. L., Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Anal. non linéaire, to appear. | Zbl 0901.35023

[15] Ding, W. Y. - Ni, W. M., On the existence of a positive entire solution of a semilinear elliptic equation. Arch. Rat. Mech. Anal., 91, 1986, 283-308. | MR 807816 | Zbl 0616.35029

[16] Esteban, M. J. - Lions, P. L., Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh, 93, 1982, 1-14. | MR 688279 | Zbl 0506.35035

[17] Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations via variational methods. Comm. in PDE, 21, 1996, 787-820. | MR 1391524 | Zbl 0857.35116

[18] Li, Y., Remarks on a semilinear elliptic equations on $R^{n}$ . J. Diff. Eq., 74, 1988, 34-39. | MR 949624 | Zbl 0662.35038

[19] Li, Y. Y., Prescribing scalar curvature on $S^{3}$ , $S^{4}$ and related problems. J. Funct. Anal., 118, 1991, 43-118. | MR 1245597 | Zbl 0790.53040

[20] Li, Y. Y., On a singularly perturbed elliptic equation. Adv. Diff. Eq., 2, 1997, 955-980. | MR 1606351 | Zbl 1023.35500

[21] Lions, P. L., The concentration-compactness principle in the calculus of variations: the locally compact case. Part I, II. Ann. Inst. H. Poincaré, Anal. non linéaire, 1, 1984, 109-145; 223-283. | Zbl 0704.49004

[22] Montecchiari, P., Multiplicity results for a class of Semilinear Elliptic Equations on $R^{m}$ . Rend. Sem. Mat. Univ. Padova, 95, 1996, 217-252. | MR 1405365 | Zbl 0866.35043

[23] Musina, R., Multiple positive solutions of the equation $- △ u - λ u + k (x) u^{p - 1} = 0$ in $R^{n}$ . Top. Meth. Nonlinear Anal., 7, 1996, 171-185. | MR 1422010 | Zbl 0909.35042

[24] Rabinowitz, P. H., A note on a semilinear elliptic equation on $R^{n}$ . In: A. Ambrosetti - A. Marino (eds.), Nonlinear Analysis, a tribute in honour of Giovanni Prodi. Quaderni della Scuola Normale Superiore, Pisa 1991. | MR 1205391 | Zbl 0836.35045

[25] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys., 43, 1992, 270-291. | MR 1162728 | Zbl 0763.35087

[26] Séré, E., Looking for the Bernoulli shift. Ann. Inst. H. Poincaré, Anal. non linéaire, 10, 1993, 561-590. | MR 1249107 | Zbl 0803.58013

[27] Strauss, W. A., Existence of solitary waves in higher dimensions. Comm. Math. Phys., 55, 1979, 149-162. | MR 454365 | Zbl 0356.35028

[28] Stuart, C. A., Bifurcation in $L^{p} (R^{n})$ for a semilinear elliptic equation. Proc. London Math. Soc., (3), 57, 1988, 511-541. | MR 960098 | Zbl 0673.35005