We review our joint result with E. Lenzmann about the uniqueness of ground state solutions of non-linear equations involving the fractional Laplacian and provide an alternate uniqueness proof for an equation related to the intermediate long-wave equation.
@incollection{JEDP_2011____A5_0, author = {Frank, Rupert L.}, title = {On the uniqueness of ground states of non-local equations}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {5}, pages = {1--10}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2011}, doi = {10.5802/jedp.77}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.77/} }
TY - JOUR AU - Frank, Rupert L. TI - On the uniqueness of ground states of non-local equations JO - Journées équations aux dérivées partielles PY - 2011 SP - 1 EP - 10 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.77/ DO - 10.5802/jedp.77 LA - en ID - JEDP_2011____A5_0 ER -
Frank, Rupert L. On the uniqueness of ground states of non-local equations. Journées équations aux dérivées partielles (2011), article no. 5, 10 p. doi : 10.5802/jedp.77. http://www.numdam.org/articles/10.5802/jedp.77/
[1] J. P. Albert, Positivity properties and uniqueness of solitary wave solutions of the intermediate long-wave equation. In: Evolution equations (Baton Rouge, LA, 1992), 11–20, Lecture Notes in Pure and Appl. Math. 168, Dekker, New York, 1995. | MR | Zbl
[2] J. P. Albert, J. L. Bona, Total positivity and the stability of internal waves in stratified fluids of finite depth. IMA J. Appl. Math. 46 (1991), no. 1-2, 1–19. | MR | Zbl
[3] J. P. Albert, J. L. Bona, J.-C. Saut, Model equations for waves in stratified fluids. Proc. R. Soc. Lond. A 453 (1997), 1233–1260. | MR | Zbl
[4] J. P. Albert, J. F. Toland, On the exact solutions of the intermediate long-wave equation. Differential Integral Equations 7 (1994), no. 3–4, 601–612. | MR | Zbl
[5] C. J. Amick, J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation – a nonlinear Neumann problem in the plane. Acta Math. 167 (1991), no. 1-2, 107–126. | MR | Zbl
[6] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2) 138 (1993), no. 1, 213–242. | MR | Zbl
[7] W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343. | MR | Zbl
[8] Ch. V. Coffman, Uniqueness of the ground state solution for and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81–95. | MR | Zbl
[9] R. L. Frank, E. Lenzmann, On ground states for the -critical boson star equation. Preprint (2009), arXiv:0910.2721.
[10] R. L. Frank, E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in , Acta Math., to appear. Preprint (2010), arXiv:1009.4042.
[11] R. L. Frank, E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality. In: Spectral Theory, Function Spaces and Inequalities, B. M. Brown et al. (eds.), 55–67, Oper. Theory Adv. Appl. 219, Birkhäuser, Basel, 2011.
[12] R. L. Frank, E. H. Lieb, R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21 (2008), no. 4, 925–950. | MR | Zbl
[13] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), 3407–3430. | MR | Zbl
[14] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. | MR | Zbl
[15] R. I. Joseph, Solitary waves in a finite depth fluid. J. Phys. A 10 (1977), L225–L227. | MR | Zbl
[16] C.E. Kenig, Y. Martel, L. Robbiano, Local well-posedness and blow up in the energy space for a class of critical dispersion generalized Benjamin–Ono equations. Ann. IHP (C) Non Linear Analysis, 28 (2011), no. 6, 853–887.
[17] M. K. Kwong, Uniqueness of positive solutions of in . Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. | MR | Zbl
[18] Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153–180. | MR | Zbl
[19] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349–374. | MR | Zbl
[20] E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. | MR | Zbl
[21] E. H. Lieb, H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112 (1987), no. 1, 147–174. | MR | Zbl
[22] L. Ma, L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455–467. | MR | Zbl
[23] K. McLeod, J. Serrin, Uniqueness of positive radial solutions of in . Arch. Rational Mech. Anal. 99 (1987), no. 2, 115–145. | MR | Zbl
Cité par Sources :