Fourth-Order Problems
with
Université Libre de Bruxelles Faculté des Sciences Département de Mathématique
Fourth-Order Problems with Mixed Dispersion
Thèse présentée en vue de l’obtention du titre de Docteur en Sciences
Robson Alves do Nascimento Filho
Promoteur de Thèse: Denis Bonheure
The work presented in this thesis was carried out at the Université Libre de Bruxelles (ULB), Department of Mathematics, in the Service de recherche analyse et équations aux dérivées partielles.
Fourth-Order Problems with Mixed Dispersion Typeset in LATEX with the KOMA-Script class.
Fourth-Order Problems with Mixed Dispersion,
Thèse présentée en vue de l’obtention du titre de Docteur en Sciences.
Jury de thèse:
Promoteur: Denis Bonheure, Université Libre de Bruxelles, Belgium
Président: Bruno Premoselli, Université Libre de Bruxelles, Belgium
Secrétaire: Jean-Baptiste Castéras, Université Libre de Bruxelles, Belgium
Filippo Gazzola, Politecnico di Milano, Italy
Ederson Moreira dos Santos, Universidade de São Paulo, Brazil
Lieu: Bruxelles
за България, за нейната история за нейната култура, за нейният народ
Acknowledgments
“No one who achieves success does so without the help of others. The wise and confident
acknowledge this help with gratitude.”
— Alfred North Whitehead “Gratitude is not only the greatest of virtues, but the parent of all the others.”
— Marcus Tullius Cicero
Over the years, many people have helped and encouraged me and thereby, directly or indirectly, played a part in this manuscript. In the next lines, I shall express my gratitude to the usual suspects: advisor, collaborators, committee, professors, friends, and family.
I wish to express my deep gratitude to Denis Bonheure who gave of his time and energy to help bring this Ph.D. thesis to its conclusion. Je tiens à lui remercier très sincèrement pour m’avoir guidé dans ce travail. Sa compétence, sa patience et sa rigueur mathématique m’ont beaucoup appris. Agradeço também pelas discussões durante les repas de midi, e pelos seminários de análise.
Je tiens à exprimer ma reconnaissance à Jean-Baptiste Castéras pour des dis-cussions mathématiques ainsi que pour l’organisation des séminaires. Agradeço ao Ederson Moreira dos Santos pelas discussões sobre simetrizações e sobre a cultura matématica. Valeu professor!
J’adresse mes plus chaleureux remerciements à Bruno Premoselli. Sa lecture im-peccable et ses suggestions ont été d’une aide inestimable. Grazie mille per tutto!
Pendant mon parcours de doctorat j’ai beaucoup apprécié les travaux de Filippo Gazzola sur les équations polyharmoniques. La clarté de ses idées m’a toujours fasciné. Il m’est donc un grand honneur qu’il ait accepté de faire partie de ce jury de thèse. Grazie mille!
Un grand merci à Jean-Pierre Gossez qui a toujours été disponible pour une bonne conversation. Je le remercie très vivement de m’avoir présenté à ses collaborateurs. En particulier, Pedro Ubilla et Djairo de Figueiredo. Sempre foi um prazer ouvir os conselhos dos mais experientes.
Durante os meus anos como estudante na Universidade de Brasília tive a oportu-nidade de aprender e de discutir matemática com várias pessoas. Sou muito grato aos professores Nigel Pitt, Mauro Rabelo, João Carlos Nascimento e Marcelo Fur-tado. Em particular, obrigado ao meu chapa Marcelão por várias discussões sobre a teoria de linking.
Over the last five years soccer has been my main tool to shape my mind. During those years, I have had the pleasure to play for Los Chorizos Biónicos. Hereby I would like to thank all my teammates. También me gustaría agradecer a mis amigos del Más Fútbol por los partidos semanales en la VUB, y por muchas discusiones fuera de la cancha. Many thanks to Andrea, Shyam, Céline and Nacho (el gran capitán). Quiero expresar mi más sincero agradecimiento a mis amigos y compañeros de batallas: Gus (el monstruo de Madrid) y Cristian (el weón). Ellos me han enseñado mucho más que la física y el castellano.
Je voudrais remercier les collègues du service analyse de l’ULB: Isabel Coelho, Manon Nys, Hussein Cheikh-Ali, Nicola Abatangelo et Alessandro Iacopetti (il più grande chitarrista italiano). Um muito obrigado a Isabel por muitas conversas ao longo dos anos.
Parmi mes collègues je remercie infiniment Ann Kiefer, Mauricio Caicedo, Rémi Dendievel, Łukasz Kidziński, Audrey Herinckx, Thomas Connor, Eglantine Camby et Marcelo Alves. Muito obrigado ao Marcelo pelas discussões sobre o fluxo de Ricci e o programa de Thurston.
A l’ULB j’ai eu l’occasion de discuter et d’acquérir une connaissance générale en mathématiques en parlant aux gens du département. Pour cela je tiens à remercier Henri Anciaux, Guillaume Dujardin et Michele D’Adderio.
Je remercie Malou et Edwine pour l’excellent service et pour l’aide sur les affaires administratives de l’ULB.
Agradeço a minha família pelo apoio durante todos esses anos. Valeu galerinha! И накрая, благодаря на Стела Аспарухова за силната и подкрепа и за това че ме научи на много неща в живота.
Contents
Introduction 1
1 Waveguide solutions for a nonlinear Schrödinger equation with mixed
dispersion 13
1.1 Introduction . . . 13
1.2 Functional framework . . . 16
1.3 Existence of minimizers . . . 19
1.4 Sign and symmetry . . . 23
1.5 The effect of a small fourth-order dissipation . . . 25
1.6 Sign-changing radial minimizer . . . 32
1.7 Further comments . . . 36
2 Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation 37 2.1 Introduction . . . 37
2.2 Existence of standing waves with a prescribed mass . . . 44
2.2.1 Gagliardo-Nirenberg interpolation inequalities . . . 45
2.2.2 Estimates of the energy . . . 46
2.2.3 Subcritical H1 exponents . . . 50
2.2.4 Subcritical H2 exponents . . . 50
2.3 Qualitative properties . . . 53
2.3.1 Existence of positive standing waves with a prescribed mass . 53 2.3.2 Radial symmetry of at least one minimal standing wave with prescribed mass . . . 56
2.3.3 Radial sign-changing minimal standing waves with prescribed mass . . . 57
2.3.4 Radial symmetry of all ground states in the strong second order dispersion case or weak fourth-order dispersion case . . . 58
2.3.5 Exponential decay . . . 62
2.3.6 Nondegeneracy in dimension one . . . 69
2.4 The effect of a small fourth-order dispersion . . . 75
2.4.1 Standing waves with a prescribed mass . . . 76
2.4.2 Ground states . . . 83
2.5 Orbital stability . . . 84
2.5.1 Standing waves with a prescribed mass . . . 86
2.5.2 Nondegenerate standing waves . . . 87
2.6 Further comments . . . 93
Contents
3 Infinitely many radial sign-changing solutions to a nonlinear Schrödinger
equation with mixed dispersion 95
3.1 Introduction . . . 95
3.2 Abstract framework . . . 97
3.2.1 Localizing critical points . . . 100
3.3 Decomposition method with respect to dual cones . . . 104
3.4 A multiplicity result . . . 112
3.5 Morse index of a radial sign-changing solution . . . 117
3.6 Further comments . . . 119
4 Existence of a nonconstant solution to a fourth-order equation with critical exponent 121 4.1 Introduction . . . 121
4.2 Preliminaries . . . 124
4.3 A relation between Σν(RN) and S . . . 127
4.4 Asymptotic estimates . . . 135
4.5 A Sobolev inequality of second order . . . 143
4.6 An inequality in H2(Ω) . . . 147
4.7 Further comments . . . 153
Bibliography 155
