On the uniqueness of ground states of non-local equations
Journées équations aux dérivées partielles, (2011), article no. 5, 10 p.

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.

@article{JEDP_2011____A5_0,
     author = {Frank, Rupert L.},
     title = {On the uniqueness of ground states of non-local equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2011},
     doi = {10.5802/jedp.77},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2011____A5_0}
}
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/item/JEDP_2011____A5_0/

[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 1300416 | Zbl 0818.35100

[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 1106250 | Zbl 0723.76106

[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 1455330 | Zbl 0886.35111

[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 1270094 | Zbl 0937.35142

[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 1111746 | Zbl 0755.35108

[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 1230930 | Zbl 0826.58042

[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 2200258 | Zbl 1093.45001

[8] Ch. V. Coffman, Uniqueness of the ground state solution for Δu-u+u 3 =0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81–95. | MR 333489 | Zbl 0249.35029

[9] R. L. Frank, E. Lenzmann, On ground states for the L 2 -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 2425175 | Zbl 1202.35146

[13] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), 3407–3430. | MR 2469027 | Zbl 1189.26031

[14] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. | MR 2360010 | Zbl 1208.65001

[15] R. I. Joseph, Solitary waves in a finite depth fluid. J. Phys. A 10 (1977), L225–L227. | MR 455822 | Zbl 0369.76018

[16] C.E. Kenig, Y. Martel, L. Robbiano, Local well-posedness and blow up in the energy space for a class of L 2 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 Δu-u+u p =0 in n . Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. | MR 969899 | Zbl 0676.35032

[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 2055032 | Zbl 1075.45006

[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 717827 | Zbl 0527.42011

[20] E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. | MR 1817225 | Zbl 0966.26002

[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 904142 | Zbl 0641.35065

[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 2592284 | Zbl 1185.35260

[23] K. McLeod, J. Serrin, Uniqueness of positive radial solutions of -Δu+f(u)=0 in R n . Arch. Rational Mech. Anal. 99 (1987), no. 2, 115–145. | MR 886933 | Zbl 0667.35023