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/

 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

 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

 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

 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

 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

 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

 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

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

 R. L. Frank, E. Lenzmann, On ground states for the ${L}^{2}$-critical boson star equation. Preprint (2009), arXiv:0910.2721.

 R. L. Frank, E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ℝ$, Acta Math., to appear. Preprint (2010), arXiv:1009.4042.

 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.

 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

 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

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

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

 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.

 M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+{u}^{p}=0$ in ${ℝ}^{n}$. Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. | MR 969899 | Zbl 0676.35032

 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

 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

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

 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

 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

 K. McLeod, J. Serrin, Uniqueness of positive radial solutions of $-\Delta u+f\left(u\right)=0$ in ${R}^{n}$. Arch. Rational Mech. Anal. 99 (1987), no. 2, 115–145. | MR 886933 | Zbl 0667.35023
{"ft_lang_label":"__label__en","ft_lang_prob":0.66248184,"math_prob":0.96242994,"size":4991,"snap":"2020-24-2020-29","text_gpt3_token_len":1747,"char_repetition_ratio":0.12993784,"word_repetition_ratio":0.03548796,"special_character_ratio":0.41134042,"punctuation_ratio":0.29411766,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97213846,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-06T15:03:42Z\",\"WARC-Record-ID\":\"<urn:uuid:125fa40d-41e9-49cb-adb6-ee7be7871e06>\",\"Content-Length\":\"33980\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8c3626bb-d908-4ca6-b3ba-2680e0ba532e>\",\"WARC-Concurrent-To\":\"<urn:uuid:42778f0b-f078-4064-9ff1-1791897bfffb>\",\"WARC-IP-Address\":\"129.88.220.36\",\"WARC-Target-URI\":\"http://archive.numdam.org/item/JEDP_2011____A5_0/\",\"WARC-Payload-Digest\":\"sha1:CPRROKEVHAAQ56W3H3NDOZZKFVEMVGCQ\",\"WARC-Block-Digest\":\"sha1:4EMOCUUPQGRE5HIUZGQHGDX3NTI4LQ4A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590348513321.91_warc_CC-MAIN-20200606124655-20200606154655-00574.warc.gz\"}"}