Chipot m quittner p (eds ) handbook of differential equations stationary partial differential equations vol 2

H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS VOLUME II This page intentionally left blank H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS Volume II Edited by M CHIPOT Institute of Mathematics, University of Zürich, Zürich, Switzerland P QUITTNER Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovak Republic 2005 ELSEVIER Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER B.V Radarweg 29 P.O Box 211 1000 AE Amsterdam The Netherlands ELSEVIER Inc 525 B Street, Suite 1900 San Diego, CA 92101-4495 USA ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR UK © 2005 Elsevier B.V All rights reserved This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions) In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance 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noted above Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made First edition 2005 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record is available from the British Library ISBN: 444 52045 Set ISBN: 444 51743 x The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands Preface This handbook is Volume II in a series devoted to stationary partial differential equations Similarly as Volume I, it is a collection of self-contained, state-of-the-art surveys written by well-known experts in the field The topics covered by this handbook include existence and multiplicity of solutions of superlinear elliptic equations, bifurcation phenomena, problems with nonlinear boundary conditions, nonconvex problems of the calculus of variations and Schrödinger operators with singular potentials We hope that these surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics We thank all the contributors for their clearly written and elegant articles We also thank Arjen Sevenster and Andy Deelen at Elsevier for efficient collaboration M Chipot and P Quittner v This page intentionally left blank List of Contributors Bartsch, T., Universität Giessen, 35392 Giessen, Germany (Ch 1) Dacorogna, B., EPFL, 1015 Lausanne, Switzerland (Ch 2) Du, Y., University of New England, Armidale, NSW 2351, Australia (Ch 3) López-Gómez, J., Universidad Complutense de Madrid, 28040 Madrid, Spain (Ch 4) Melgaard, M., Uppsala University, S-751 06 Uppsala, Sweden (Ch 6) Rossi, J.D., Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (Ch 5) Rozenblum, G., Chalmers University of Technology, and University of Gothenburg, S-412 96 Gothenburg, Sweden (Ch 6) Solimini, S., Politecnico di Bari, 70125 Bari, Italy (Ch 7) Wang, Z.-Q., Utah State University, Logan, UT 84322, USA (Ch 1) Willem, M., Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (Ch 1) vii This page intentionally left blank Contents Preface List of Contributors Contents of Volume I v vii xi The Dirichlet Problem for Superlinear Elliptic Equations T Bartsch, Z.-Q Wang and M Willem Nonconvex Problems of the Calculus of Variations and Differential Inclusions B Dacorogna Bifurcation and Related Topics in Elliptic Problems Y Du Metasolutions: Malthus versus Verhulst in Population Dynamics A Dream of Volterra J López-Gómez Elliptic Problems with Nonlinear Boundary Conditions and the Sobolev Trace Theorem J.D Rossi Schrödinger Operators with Singular Potentials G Rozenblum and M Melgaard Multiplicity Techniques for Problems without Compactness S Solimini 57 127 211 311 407 519 Author Index 601 Subject Index 609 ix 598 S Solimini Acknowledgment The author is grateful to Francesco Maddalena for his continuous support in organizing notes provided in the Introduction References [1] A Ambrosetti, M Badiale and S Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch Ration Mech Anal 140 (1997), 285–300 [2] F.V Atkinson, H Brezis and L.A Peletier, Solutions d’Equations elliptiques avec exposant de Sobolev critique qui changent de signe, C R Math Acad Sci Paris 306 (1988), 711–714 [3] F.V Atkinson, H Brezis and L.A Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J Differential Equations 85 (1990), 151–170 [4] A Bahri and P.L Lions, 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 [5] T Bartsch and Z.Q Wang, Sign changing solutions of nonlinear Schrödinger equations, Topol Methods Nonlinear Anal 13 (1999), 191–198 [6] T Bartsch and M Willem, Infinitely many nonradial solutions of an Euclidean scalar field equation, J Funct Anal 117 (1993), 447–460 [7] V Benci and G Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch Ration Mech Anal 99 (1987), 283–300 [8] H Berestycki and P.L Lions, Nonlinear scalar field equations, I Existence of a ground state, II Existence of infinitely many solutions, Arch Ration Mech Anal 82 (1983), 313–346, 347–376 [9] M.S Berger, On the existence and structure of stationary states for a nonlinear Klein–Gordon equation, J Funct Anal (1972), 249–261 [10] H Brezis and T Kato, Remarks on the Schrödinger operator with singular complex potential, J Math Pures Appl 58 (1979), 137–151 [11] H Brezis and L Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm Pure Appl Math 36 (1983), 437–477 [12] A Capozzi, D Fortunato and G Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann Inst H Poincaré Anal Non Linéaire (1985), 463–470 [13] G Cerami, G Devillanova and S Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc Var Partial Differential Equations 23 (2005), 139–168 [14] G Cerami, S Solimini and M Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J Funct Anal 69 (1986), 289–306 [15] M Clapp and T Weth, Multiple solutions for the Brezis–Nirenberg problem, Preprint [16] C.V Coffman, Uniqueness of the ground state solution for − u − u + u3 = and a variational characterization of other solutions, Arch Ration Mech Anal 46 (1972), 81–95 [17] S Coleman, V Glaser and A Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm Math Phys 58 (1978), 211–221 [18] V Coti Zelati and P Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on RN , Comm Pure Appl Math 10 (1992), 1217–1269 [19] G Devillanova and S Solimini, Concentrations estimates and multiple solutions to elliptic problems at critical growth, Adv Differential Equations (2002), 1257–1280 [20] G Devillanova and S Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comun Contemp Math (2003), 171–177 [21] W.-Y Ding and W.-M Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch Ration Mech Anal 91 (1986), 283–308 [22] J Dugundji, Topology, 8th Edition, Allin and Bacon, Boston, MA (1973) [23] A Floer and A Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J Funct Anal 69 (1986), 397–408 Multiplicity techniques for problems without compactness 599 [24] D Fortunato and E Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc Roy Soc Edinburgh Sect A 105 (1987), 205–213 [25] E Jannelli and S Solimini, Concentration estimates for critical problems, Ricerche Mat Sect A XLVIII (1999), Supplemento, 233–257 [26] M.A Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York (1964) [27] A Lazer and S Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min–max type, Nonlinear Anal 12 (1988), 761–775 [28] P.L Lions, The concentration compactness principle in the calculus of variations, Parts I and II, Ann Inst H Poincaré Anal Non Linéaire (1984), 109–145, 223–283 [29] P.L Lions, The concentration–compactness principle in the calculus of variations The limit case – Part II, Rev Mat Iberoamericana 12 (1985), 45–121 [30] A Manes and A.M Michelettii, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll Unione Mat Ital (1973), 285–301 [31] C Maniscalco, Multiple solutions for semilinear elliptic problems in RN , Ann Univ Ferrara 37 (1991), 95–110 [32] A Marino and G Prodi, Metodi perturbativi della teoria di Morse, Boll Unione Mat Ital 11 (1975), 1–32 [33] C Miranda, Un’osservazione sul teorema di Brower, Boll Unione Mat Ital Sez II XIX (1940), 5–7 [34] R Molle, M Musso and D Passaseo, Positive solutions for a class of nonlinear elliptic problems in RN , Proc Roy Soc Edinburgh Sect A 130 (2000), 141–166 [35] Z Nehari, On a nonlinear differential equation arising in nuclear physics, Proc Roy Irish Acad 62 (1963), 117–135 [36] Y.G Oh, On positive multi bump bound states of nonlinear Schrödinger equations under multiple wall potential, Comm Math Phys 131 (1990), 223–253 [37] S Pohozaev, Eigenfunctions of the equation u + λf (u) = 0, Soviet Mat Dokl (1965), 1408–1411 [38] P.H Rabinowitz, On a class of nonlinear Schrödinger equations, Z Angew Math Phys 43 (1992), 270–291 [39] S Solimini, On the existence of infinitely many radial solutions for some elliptic problems, Rev Mat Apl (1986), 75–86 [40] S Solimini, Morse index estimates in min–max theorems, Manuscripta Math 63 (1989), 421–453 [41] S Solimini, Lecture notes on some estimates in functional analysis via convolution techniques, Internal Report 91, 1, I.R.M.A – Bari, Italy (1991) [42] S Solimini, A note on compactness-type properties with respect to Lorentz norm of bounded subsets of a Sobolev spaces, Ann Inst H Poincaré Anal Non Linéaire 12 (1995), 319–337 [43] S Solimini, Min–Max levels on the double natural constraint, Preprint [44] S Solimini, Notes on Min–Max Theorems, Lecture Notes SISSA, Trieste, Italy (1989) [45] M Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math Z 187 (1984), 511–517 [46] M Struwe, Variational Methods: Applications to Nonlinear PDE & Hamiltonian Systems, Springer-Verlag, Berlin (1990) This page intentionally left blank Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned Italic numbers refer to reference pages Numbers between brackets are the reference numbers No distinction is made between first and co-author(s) Abreu, E 315, 375, 401 [1] Acerbi, E 73, 123 [1] Adam, C 491, 512 [1]; 512 [2] Adami, R 421, 512 [3] Adams, D 433, 436, 441, 442, 512 [4] Adams, R.A 316, 402 [2] Adimurthi 344, 361, 368, 369, 402 [3] Aftalion, A 31, 33, 52 [1] Aharonov, Y 421, 487, 512 [5]; 512 [6] Ahmedou, M.O 314, 344, 346, 404 [60]; 404 [61] Aizenman, M 469, 512 [7] Akilov, G.P 460, 514 [60] Ali, S.W 169, 205 [AC] Alibert, J.J 65, 123 [2] Allaire, G 62, 123 [3] Amann, H 129, 147, 156, 166, 205 [Am]; 226, 305, 306 [1]; 306 [2]; 306 [3]; 314, 321, 402 [4] Ambrosetti, A 3, 5, 31, 33, 52 [2]; 52 [3]; 52 [4]; 52 [5]; 298, 306 [4]; 314, 325, 326, 344, 346, 402 [5]; 402 [6]; 526, 598 [1] Anane, A 351, 402 [7] Andreu, F 315, 370, 371, 396, 402 [8]; 402 [9]; 402 [10]; 402 [11]; 402 [12] Arcoya, D 315, 321, 395, 396, 399, 400, 402 [13]; 402 [14] Arendt, W 460, 512 [8] Aris, R 192, 206 [Ar] Atkinson, F.V 522–524, 594, 598 [2]; 598 [3] Aubert, G 61, 123 [4]; 123 [5]; 124 [6] Aubin, T 313, 369, 402 [15] Avron, J 474, 482, 492, 494, 512 [9] Ball, J.M 63, 121, 124 [7]; 124 [8] Ballester, C 371, 402 [8] Bandle, C 218, 301, 303, 306 [5]; 306 [6]; 306 [7] Bandyopadhyay, S 81, 124 [9] Barles, G 313, 402 [16] Barroso, A.C 81, 124 [9] Bartsch, T 17, 24, 27, 29, 31, 33, 34, 38, 40–42, 51, 52 [10]; 52 [11]; 52 [12]; 52 [13]; 52 [14]; 52 [15]; 52 [16]; 52 [17]; 52 [18]; 52 [19]; 52 [20]; 387, 391, 393, 402 [17]; 525, 526, 598 [5]; 598 [6] Bauman, P 61, 124 [10] Bebernes, J 170, 206 [BE] Beckner, W 344, 402 [18] Benci, V 17, 52 [21]; 52 [22]; 52 [23]; 525, 527, 533, 598 [7] Benguria, R 472, 512 [11] Bénilan, Ph 315, 396, 402 [19] Berestycki, H 18, 20, 35, 40, 52 [7]; 52 [24]; 52 [25]; 53 [26]; 525, 598 [8] Berezanski, Yu 426, 512 [12] Berezin, F 409, 420, 430, 446, 512 [13] Berger, M.S 525, 598 [9] Biezuner, R.J 369, 402 [20] Birman, M.Sh 409, 412, 414, 445–447, 456, 464, 467, 507, 510, 511, 512 [14]; 512 [15]; 512 [16]; 512 [17]; 512 [18] Blanchard, Ph 472, 512 [19] Blat, J 150, 163, 169, 206 [BB1]; 206 [BB2] Bohm, D 421, 512 [5] Bolle, Ph 35, 53 [27]; 53 [28] Bony, J.M 139, 206 [Bo] Borzov, V.V 412, 512 [15] Bourgain, J 318, 402 [21] Braverman, M 421, 428, 431, 432, 512 [20] Bressan, A 74, 81, 124 [11] Badiale, M 526, 598 [1] Bahri, A 8, 18, 35, 52 [6]; 52 [7]; 52 [8]; 52 [9]; 525, 526, 533, 598 [4] Balinsky, A 491, 512 [10] 601 602 Author Index Brezis, H 7, 8, 12, 18, 20, 21, 27, 52 [3]; 53 [29]; 53 [30]; 53 [31]; 53 [32]; 53 [33]; 297, 299, 306 [8]; 318, 330, 339, 367, 368, 402 [21]; 402 [22]; 522–524, 546, 555, 557, 594, 598 [2]; 598 [3]; 598 [10]; 598 [11] Brock, F 9, 17, 53 [34]; 315, 395, 402 [23] Browder, F 299, 306 [9] Brown, K.J 150, 153, 163, 169, 170, 206 [BB1]; 206 [BB2]; 206 [BIS]; 207 [DB] Bugliaro, L 499, 500, 502, 512 [21]; 513 [22] Bukhvalov, A.V 460, 512 [8] Buttazzo, G 61, 69, 124 [12]; 124 [13] Byeon, J 12, 53 [35] Cabré, X 314, 344, 402 [24] Calderon, A.P 317, 402 [25] Cano-Casanova, S 213, 299, 304, 306 [10]; 306 [11]; 306 [12]; 307 [13] Cantrell, R.S 169, 206 [CC]; 206 [CCH] Capozzi, A 522, 598 [12] Carriao, P 315, 375, 401 [1] Caselles, V 371, 402 [8] Casher, A 487, 512 [6] Castro, A 31, 53 [36]; 53 [37]; 170, 206 [CS] Catrina, F 12, 53 [38] Celada, P 61, 124 [14]; 124 [15] Cellina, A 61, 62, 74, 81, 97, 114, 124 [16]; 124 [17]; 124 [18]; 124 [19] Cerami, G 17, 52 [3]; 52 [21]; 52 [22]; 52 [23]; 521, 523, 525–527, 533, 581, 594, 598 [7]; 598 [13]; 598 [14] Cesari, L 61, 124 [20]; 124 [21] Chabrowski, J 3, 53 [39]; 53 [40]; 336, 375, 402 [26]; 402 [27] Chang, K.C 3, 30, 53 [41] Chanillo, S 465, 513 [23] Chavel, I 466, 513 [24] Cherkaev, A 365, 402 [28] Cherkaeva, E 365, 402 [28] Chernoff, P.R 427, 513 [25] Cherrier, P 313, 320, 321, 356, 402 [29] Chipot, M 314, 341–343, 403 [30]; 403 [31]; 403 [32]; 403 [33] Chlebík, M 314, 315, 343, 368, 403 [30]; 403 [34]; 403 [35] Cingolani, S 526, 598 [1] Cỵrstea, F C., St ¸ 302, 303, 307 [14]; 307 [15]; 307 [16]; 307 [17]; 342, 403 [36]; 403 [37] Clapp, M 523, 594, 598 [15] Clement, P 395, 396, 403 [38] Cobos, F 473, 513 [26] Coffman, C.V 11, 12, 53 [42]; 53 [43]; 525, 598 [16] Coleman, S 525, 598 [17] Colombo, G 61, 124 [18] Consul, N 314, 403 [39] Conti, M 27, 31, 53 [44]; 53 [45] Conway, E 199, 206 [CHS] Coron, J.-M 8, 18, 52 [8] Cortázar, C 398, 403 [40] Cosner, C 169, 205 [AC]; 206 [CC]; 206 [CCH] Cossio, J 31, 53 [36]; 53 [37] Costa, D.G 16, 34, 53 [46]; 53 [47] Coti Zelati, V 36, 45–49, 53 [48]; 526, 598 [18] Courant, R 301, 307 [18]; 409, 513 [27] Crandall, M.G 130, 131, 177, 189, 206 [CR1]; 206 [CR2]; 237, 239, 307 [19]; 307 [20]; 315, 396, 402 [19] Cutrì, A 61, 124 [22] Cwikel, M 452, 455, 513 [28] Cycon, H.L 409, 425, 492, 513 [29] Dacorogna, B 61, 62, 64, 65, 67–69, 72–74, 76–79, 81–84, 86, 91, 92, 94, 97, 105, 109, 111, 113–117, 121–123, 123 [2]; 124 [9]; 124 [12]; 124 [23]; 124 [24]; 124 [25]; 124 [26]; 124 [27]; 124 [28]; 124 [29]; 124 [30]; 124 [31]; 124 [32]; 124 [33]; 125 [34]; 125 [35]; 125 [36]; 125 [37] Dancer, E.N 31, 53 [49]; 53 [50]; 144, 162, 165, 168–171, 189, 191–193, 200, 205, 206 [DD1]; 206 [DD2]; 206 [DLO]; 206 [Da1]; 206 [Da2]; 206 [Da3]; 206 [Da4]; 206 [Da5]; 206 [Da6]; 206 [Da7]; 206 [Da8]; 299, 304, 307 [21]; 307 [22] Davila, J 315, 344, 395, 397, 398, 403 [41]; 403 [42] De Blasi, F.S 74, 81, 125 [38] de Figueiredo, D.G 21, 53 [51]; 314, 378, 379, 385–387, 391, 393, 402 [17]; 403 [43]; 403 [44]; 403 [45] De la Bretéche, R 470, 513 [30] de Mottoni, P 205, 206 [dMR] Deimling, K 129, 156, 206 [De] del Pino, M 144, 206 [dP]; 298, 307 [26]; 315, 347, 356, 360, 361, 403 [46] Delgado, M 303, 304, 307 [23]; 307 [24]; 307 [25] Demengel, F 315, 403 [47] Denzler, J 364–366, 403 [48] Devillanova, G 521, 523, 524, 526, 532, 594, 598 [13]; 598 [19]; 598 [20] Ding, W.-Y 526, 598 [21] Ding, Y 51, 52 [11] Drábek, P 358, 399, 403 [49] Druet, O 313, 403 [50] Du, Y 31, 53 [49]; 53 [50]; 133, 137, 142, 144, 147, 150, 153, 158, 163, 164, 167–171, 189, Author Index 192, 200, 206 [DD1]; 206 [DD2]; 206 [Du1]; 206 [Du2]; 207 [DB]; 207 [DG]; 207 [DH]; 207 [DHs]; 207 [DL]; 207 [DLo1]; 207 [DLo2]; 207 [DM]; 207 [DO]; 207 [Du3]; 207 [Du4]; 207 [Du5]; 207 [Du6]; 301–304, 307 [21]; 307 [27]; 307 [28] Dufresnoy, A 450, 513 [31] Dugundji, J 573, 585, 598 [22] Eberly, D 170, 206 [BE] Ebmeyer, C 319, 403 [51] Edmunds, D.E 409, 412, 441, 513 [32] Ekeland, I 62, 125 [39] El Hamidi, A 341, 403 [52] Elgueta, M 398, 403 [40] Elton, D 491, 513 [33] Erdös, L 478–481, 489–491, 498–502, 513 [34]; 513 [35]; 513 [36]; 513 [37]; 513 [38]; 513 [39]; 513 [40]; 513 [41] Escobar, J.F 313, 314, 317, 344, 403 [53]; 403 [54]; 403 [55]; 404 [56]; 404 [57]; 404 [58]; 404 [59] Evans, W.D 409, 412, 441, 491, 512 [10]; 513 [32] Fefferman, C 453, 463, 465, 499, 500, 502, 512 [21]; 513 [22]; 513 [42]; 513 [43] Felli, V 314, 344, 346, 404 [60]; 404 [61] Felmer, P 385–387, 398, 403 [40]; 403 [45]; 404 [62]; 404 [63] Fernández Bonder, J 313–315, 317, 318, 322, 328–330, 347, 348, 350, 358, 360, 361, 365, 369, 370, 376, 379, 384, 390, 391, 404 [64]; 404 [65]; 404 [66]; 404 [67]; 404 [68]; 404 [69]; 404 [70]; 404 [71]; 404 [72]; 404 [73]; 404 [74]; 404 [75]; 404 [76] Ferone, V 61, 124 [13] Ferreira, R 315, 376, 404 [64] Fila, M 314, 315, 341, 343, 368, 403 [30]; 403 [31]; 403 [33]; 403 [34]; 403 [35]; 404 [77] Filo, J 314, 404 [77] Floer, A 526, 598 [23] Flores, C 315, 347, 356, 360, 361, 403 [46] Flores, F 74, 81, 124 [11] Fortunato, D 522, 523, 594, 598 [12]; 599 [24] Fraile, J.M 144, 207 [FKLM]; 298, 299, 307 [29] Francfort, G 62, 123 [3] Friedrichs, K 445, 449, 513 [44] Friesecke, G 62, 74, 81, 97, 125 [40] Froese, R.G 409, 425, 492, 513 [29] Fröhlich, J 499, 500, 512 [21] Fry, M 486, 513 [45]; 513 [46]; 513 [47] Furter, J.E 307 [30] 603 Fusco, N 62, 73, 123 [1]; 125 [41] Galaktionov, V 397, 404 [78] Gámez, J.L 298, 306 [4]; 396, 399, 400, 402 [13] Gangbo, W 69, 124 [12] García-Azorero, J 52 [4]; 314, 319, 322, 330, 339, 340, 350, 404 [79]; 404 [80]; 404 [81] García-Melián, J 144, 207 [GGLS]; 300–303, 307 [31]; 307 [32] Geyler, V 490, 513 [48] Ghoussoub, N 3, 35, 53 [28]; 53 [52] Giachetti, D 62, 125 [42] Gidas, B 10, 18, 21, 53 [53]; 54 [54]; 170, 172, 179, 183, 207 [GNN]; 361, 379, 382, 404 [82]; 404 [83] Gilbarg, D 129, 140, 207 [GT]; 240, 307 [33]; 321, 380, 384, 388, 405 [84] Glaser, V 469, 514 [49]; 525, 598 [17] Gómez-Rasco, R 144, 207 [GGLS]; 300–302, 307 [31]; 307 [34]; 307 [35]; 307 [36] Graf, G.M 499, 500, 502, 512 [21]; 513 [22] Grigor’yan, A 466, 514 [50] Grishanov, E 490, 513 [48] Gromov, M 123, 125 [43] Grosse, H 469, 514 [49] Guo, Z.M 144, 207 [DG] Hadamard, J 301, 307 [37] Han, Z.C 344, 346, 405 [85]; 405 [86] Hastings, S.P 170, 192, 207 [HM] Healey, T.J 193, 207 [HK] Hebey, E 313, 403 [50] Hedberg, L 433, 436, 441, 442, 512 [4] Helffer, B 425, 470, 514 [51]; 514 [52] Herbst, I 474, 482, 492, 494, 512 [9] Hess, H 431, 474, 514 [53] Hess, P 169, 207 [He] Hilbert, D 301, 307 [18]; 409, 513 [27] Hirsch, M 155, 207 [Hir] Hislop, P.D 409, 514 [54] Hoff, D 199, 206 [CHS] Horn, R.A 77, 113, 118, 125 [44]; 125 [45] Hsu, S.B 168, 205, 207 [DHs]; 207 [HSW]; 207 [Hs] Hu, B 314, 343, 379, 384, 405 [87]; 405 [88] Huang, Q 133, 144, 207 [DH]; 301–303, 307 [28] Hundertmark, D 463, 469, 470, 472, 514 [55]; 514 [56]; 514 [57] Hutson, V 169, 206 [CCH]; 207 [HLM]; 207 [HLMP]; 207 [HMP] Ibrahim, M.M.A 170, 206 [BIS] Igbida, N 396, 402 [9] Ishii, H 313, 405 [89] 604 Author Index Ivrii, V 409, 514 [58] Iwatsuka, A 490, 514 [59] James, R.D 121, 124 [8] Jannelli, E 523, 557–559, 594, 599 [24]; 599 [25] Jeanjean, L 40, 54 [55] Johnson, C.A 77, 113, 118, 125 [44]; 125 [45] Joseph, D.D 171, 191, 208 [JL] Kan-on, Y 169, 200, 208 [KY]; 208 [Ka] Kantorovich, L.V 460, 514 [60] Kato, T 27, 53 [30]; 146, 208 [Kato]; 409, 418, 428, 429, 431, 474, 514 [61]; 514 [62]; 514 [63]; 514 [64]; 524, 546, 555, 557, 598 [10] Kavian, O 3, 54 [56] Kawasaki, K 169, 208 [MK] Kawohl, B 61, 124 [13]; 364–366, 405 [90] Keller, J.B 218, 235, 301, 307 [38] Kielhöfer, H 3, 54 [57]; 193, 207 [HK]; 208 [K] Kirchheim, B 79, 84, 125 [46] Kirsch, W 409, 425, 492, 513 [29] Kishimoto, K 169, 200, 208 [KW] Klötzler, R 62, 125 [47] Koch-Medina, P 144, 207 [FKLM]; 298, 299, 307 [29] Kohn, R.V 68, 70, 116, 117, 125 [48] Koman, P 169, 208 [KL] Kondratiev, V.A 302, 307 [39]; 433, 450, 451, 514 [65]; 514 [66] Korman, P 170, 208 [KLi] Krasnosel’ski˘ı, M.A 129, 130, 208 [Kr]; 325, 330, 405 [91]; 571, 599 [26] Kryzewski, W 51, 54 [58] Kühn, T 473, 513 [26] Kwong, M.R 10, 11, 54 [59] Lami Dozo, E 315, 362–364, 369, 404 [65]; 405 [92] Laptev, A 470–472, 514 [56]; 514 [67]; 514 [68]; 514 [69]; 514 [70] Lazer, A 584, 587, 599 [27] Le Dret, H 69, 115, 125 [49] Leinfelder, H 415, 417, 432, 514 [71]; 514 [72] Leschke, H 479, 480, 514 [73] Letelier-Albornoz, R 302, 303, 307 [32] Leung, A.W 169, 208 [KL] Levin, D 459, 514 [74] Li, P 453, 458, 514 [75] Li, S 315, 388, 390, 406 [125] Li, S.J 33, 34, 54 [60]; 54 [61]; 144, 207 [DL] Li, Y 34, 54 [63]; 170, 208 [KLi]; 314, 343, 369, 405 [93]; 405 [94] Li, Y.Y 12, 54 [62]; 314, 344, 346, 402 [5]; 405 [85]; 405 [86] Lieb, E.H 7, 53 [31]; 409, 452, 454, 456, 457, 462, 468–472, 482, 486, 492, 494, 497, 499, 500, 509, 512 [7]; 514 [57]; 515 [76]; 515 [77]; 515 [78]; 515 [79]; 515 [80]; 515 [81]; 515 [82]; 515 [83]; 515 [84]; 515 [85] Lin, C.S 172, 183, 208 [LN] Lindqvist, P 352, 405 [95] Lions, P.-L 21, 35, 36, 39, 40, 52 [9]; 52 [25]; 53 [51]; 54 [64]; 139, 208 [L]; 313, 330, 331, 340, 368, 394, 405 [89]; 405 [96]; 405 [97]; 405 [98]; 525–527, 533, 598 [4]; 598 [8]; 599 [28]; 599 [29] Liu, Z 24, 27, 34, 38, 42, 52 [12]; 54 [63]; 54 [65] Liu, Z.-L 16, 24, 27, 31, 36, 38, 39, 43, 45–47, 49–51, 54 [66]; 54 [67]; 54 [68]; 54 [69] Loewner, C 302, 308 [40] Lopes, O 54 [70] López-Gómez, J 144, 169, 193, 205, 206 [DLO]; 207 [FKLM]; 207 [GGLS]; 208 [LP1]; 208 [LP2]; 208 [LS]; 208 [Lop1]; 208 [Lop2]; 213, 218, 226, 252, 257, 298–305, 306 [3]; 306 [10]; 306 [11]; 306 [12]; 307 [13]; 307 [22]; 307 [23]; 307 [24]; 307 [25]; 307 [29]; 307 [30]; 307 [31]; 307 [35]; 307 [36]; 308 [41]; 308 [42]; 308 [43]; 308 [44]; 308 [45]; 308 [46]; 308 [47]; 308 [48]; 308 [49]; 308 [50]; 308 [51]; 308 [52]; 308 [53]; 308 [54]; 308 [55]; 308 [56] Lorca, S 40, 54 [71] Loss, M 409, 472, 478, 480, 486, 491, 499, 500, 512 [11]; 515 [79]; 515 [80]; 515 [86]; 515 [87] Lou, Y 169–171, 189, 207 [DLo1]; 207 [DLo2]; 207 [HLM]; 207 [HLMP]; 208 [LN1]; 208 [LN2] Lundgren, T.S 171, 191, 208 [JL] Ma, L 137, 207 [DM] Magalhaes, C.A 16, 53 [46] Malchiodi, A 314, 344, 346, 402 [5] Mancebo, F.J 313, 405 [99] Manes, A 591, 599 [30] Maniscalco, C 599 [31] Marcellini, P 61, 62, 74, 76–79, 81, 83, 86, 91, 92, 94, 97, 106, 109, 114–117, 121–123, 124 [27]; 124 [28]; 124 [29]; 124 [30]; 124 [31]; 124 [32]; 125 [41]; 125 [50]; 125 [51]; 125 [52] Marcus, M 145, 208 [MV]; 218, 301, 303, 306 [5]; 306 [6]; 306 [7]; 308 [57]; 308 [58] Marino, A 587, 599 [32] Marshall, A.W 77, 125 [53] Martin, A 412, 469, 514 [49]; 515 [88]; 525, 598 [17] Author Index Martinez, S 315, 318, 351, 356, 360, 362, 365, 371, 376, 399, 400, 404 [66]; 404 [67]; 405 [100]; 405 [101]; 405 [102] Mascolo, E 62, 114, 125 [54]; 125 [55]; 125 [56]; 125 [57] Masuda, K 474, 514 [64] Matano, H 155, 169, 208 [MM]; 208 [Ma] Matias, J 81, 124 [9] Mawhin, J 308 [59]; 308 [60] Mazón, J.M 315, 370, 371, 396, 402 [8]; 402 [9]; 402 [10]; 402 [11]; 402 [12] Maz’ya, V.G 423, 428, 433, 435, 436, 438, 439, 441, 442, 444, 446, 447, 450, 451, 514 [65]; 515 [89]; 515 [90]; 515 [91]; 515 [92]; 515 [93]; 515 [94]; 515 [95]; 515 [96]; 515 [97] McLeod, J.B 170, 192, 207 [HM] Melgaard, M 464, 465, 476, 482, 483, 498, 515 [98]; 515 [99]; 515 [100] Merino, S 144, 207 [FKLM]; 298, 299, 307 [29] Merizzi, L 27, 53 [44] Michelettii, A.M 591, 599 [30] Milatovic, O 421, 428, 431, 432, 512 [20] Mimura, M 169, 170, 208 [MK]; 208 [MM]; 208 [MS]; 208 [Mi] Miranda, C 526, 575, 599 [33] Mironescu, P 318, 402 [21] Mischaikow, K 169, 207 [HLM]; 207 [HLMP]; 207 [HMP] Miyagaki, O 315, 375, 401 [1] Mohammed, A 494, 515 [101] Molchanov, A 449, 516 [102] Molina-Meyer, M 218, 226, 299, 304, 306 [12]; 307 [13]; 308 [53]; 308 [54] Moll, S 371, 402 [10] Molle, R 526, 599 [34] Montefusco, E 54 [72]; 342, 405 [103] Monteiro Marques, M.D.P 62, 125 [58] Montenegro, M 315, 395, 403 [41] Morrey, C.B 63, 125 [59]; 125 [60] Motron, M 315, 370, 372, 375, 405 [104] Müller, S 62, 74, 79, 91, 123, 125 [61]; 125 [62]; 126 [63] Müller-Pfeiffer, E 27, 54 [73] Muratori, B 491, 512 [1]; 512 [2] Musso, M 526, 599 [34] Nagasaki, K 191, 208 [NS] Nakashima, K 193, 197, 208 [N] Nash, C 491, 512 [1]; 512 [2] Nehari, Z 54 [74]; 525, 599 [35] Neuberger, J.M 31, 53 [36]; 53 [37] Ni, W.-M 10, 18, 53 [53]; 54 [75]; 169, 170, 172, 179, 183, 207 [GNN]; 208 [LN]; 208 [LN1]; 605 208 [LN2]; 208 [NN]; 361, 404 [82]; 526, 598 [21] Nikishin, V.A 302, 307 [39] Nirenberg, L 8, 10, 12, 18, 53 [26]; 53 [32]; 53 [53]; 170, 172, 179, 183, 207 [GNN]; 302, 308 [40]; 330, 339, 361, 367, 368, 402 [22]; 404 [82]; 522, 598 [11] Nourrigat, J 425, 514 [51] Nussbaum, R.D 21, 53 [51]; 179, 208 [NN] Oh, Y.G 526, 599 [36] Oleinik, I.M 430, 431, 516 [103] Olkin, I 77, 125 [53] Ornelas, A 62, 125 [41]; 125 [58]; 126 [64] Ortega, R 193, 205, 206 [DLO] Osserman, R 218, 235, 301, 308 [61] Oswald, L 297, 299, 306 [8] Otelbaev, M 428, 515 [92] Ouhabaz, E.-M 475, 476, 483, 515 [98]; 516 [104] Ouyang, T 144, 170, 193, 207 [DO]; 208 [Ou]; 209 [OS1]; 209 [OS2]; 298, 308 [62] Oxtoby, J.C 84, 126 [65] Pacella, F 17, 31, 33, 52 [1]; 54 [76]; 340, 405 [98] Palais, R.S 54 [77] Palmieri, G 522, 598 [12] Pankov, A 41, 52 [13] Pao, C.V 169, 209 [Pao] Pardo, R.M 205, 208 [LP1]; 208 [LP2] Park, Y.J 313, 314, 344, 405 [105]; 405 [106] Parter, S.V 170, 209 [Pa] Passaseo, D 17, 52 [23]; 526, 599 [34] Peletier, L.A 395, 396, 403 [38]; 522–524, 594, 598 [2]; 598 [3] Peller, V.V 473, 516 [105] Peral, I 52 [4]; 314, 319, 322, 330, 339, 340, 350, 404 [79]; 404 [80]; 404 [81] Perrotta, S 61, 124 [14]; 124 [15] Pflüger, K 339, 405 [107] Phillips, D 61, 124 [10] Pianigiani, G 74, 81, 125 [38] Pierrotti, D 315, 339, 375, 405 [108]; 405 [109] Pinasco, J.P 315, 390, 404 [68]; 404 [69] Pisante, G 62, 69, 79, 84, 86, 92, 111, 113, 117, 124 [33]; 125 [34]; 126 [66] Pitt, L.D 443, 473, 516 [106] Pohozaev, S 6, 54 [78]; 522, 541, 599 [37] Polacik, P 169, 207 [HLMP]; 207 [HMP] Povzner, A.Ya 426, 427, 516 [107] Prodi, G 587, 599 [32] Protter, M.H 129, 195, 197, 209 [PW]; 301, 308 [63] 606 Author Index Quittner, P 305, 308 [55]; 308 [64]; 314, 341, 342, 403 [31]; 403 [32] Rabinowitz, P.H 3, 5, 31, 33, 36, 45–49, 52 [5]; 53 [48]; 54 [79]; 129–131, 177, 189, 193, 206 [CR1]; 206 [CR2]; 209 [Ra]; 237, 239, 307 [19]; 307 [20]; 325, 326, 329, 390, 402 [6]; 405 [110]; 526, 598 [18]; 599 [38] R˘adulescu, V.D 302, 303, 307 [14]; 307 [15]; 307 [16]; 307 [17]; 342, 403 [36]; 403 [37]; 405 [103]; 494, 515 [101] Raikov, G.D 494, 498, 515 [101]; 516 [108]; 516 [109]; 516 [110]; 516 [111] Ramos, M 54 [80] Raoult, A 69, 115, 125 [49] Raymond, J.P 62, 126 [67]; 126 [68]; 126 [69]; 126 [70] Reed, M 409, 414, 417–420, 422, 423, 428, 429, 446, 456, 460, 490, 516 [112]; 516 [113]; 516 [114] Reichel, W 315, 368, 403 [35]; 405 [111] Ribeiro, A.M 62, 69, 76, 82, 86, 92, 111, 113, 114, 117, 125 [34]; 125 [35] Robert, D 470, 514 [52] Robinson, S.B 358, 399, 403 [49] Rockafellar, R.T 82, 96, 126 [71] Rofe-Beketov, F.S 430, 516 [115] Rossi, J.D 313–315, 317–319, 321, 322, 328–330, 339, 344, 347, 348, 350, 351, 356, 358, 360–362, 365, 369–371, 376, 379, 384, 390, 391, 395, 397–400, 402 [11]; 402 [14]; 403 [42]; 404 [64]; 404 [65]; 404 [66]; 404 [67]; 404 [68]; 404 [69]; 404 [70]; 404 [71]; 404 [72]; 404 [73]; 404 [74]; 404 [75]; 404 [76]; 404 [81]; 405 [100]; 405 [101]; 405 [102] Rothe, F 205, 206 [dMR] Rozenblum, G 409, 410, 412, 428, 452, 462–465, 476, 482–485, 498, 515 [98]; 515 [99]; 515 [100]; 516 [116]; 516 [117]; 516 [118]; 516 [119]; 516 [120]; 516 [121] Ruder, R 479, 480, 514 [73] Ruijsenaars, S 421, 516 [122] Sabina de Lis, J.C 144, 169, 207 [GGLS]; 208 [LS]; 300–303, 307 [31]; 307 [32]; 308 [56] Sacks, P 315, 396, 402 [19] Safarov, Yu 409, 516 [123] Sakamoto, K 170, 208 [MS] Sattinger, D.H 132, 209 [Sa]; 232, 308 [65] Schechter, M 3, 16, 54 [81]; 55 [82]; 315, 388, 390, 406 [112] Schianchi, R 62, 114, 125 [42]; 125 [55]; 125 [56]; 125 [57] Schrader, R 431, 474, 514 [53] Schwinger, J 446, 516 [124] Segura de Leon, S 315, 396, 402 [12] Shafrir, I 314, 343, 403 [30]; 403 [33] Shaposhnikova, T.O 433, 436, 441, 442, 515 [93] Shen, Z 499, 500, 516 [125]; 516 [126] Shi, J 170, 193, 209 [OS1]; 209 [OS2]; 209 [S] Shigekawa, I 490, 516 [127] Shivaji, R 170, 206 [BIS]; 206 [CS]; 209 [Sh] Shubin, M 409, 410, 412, 420, 421, 425–428, 430–433, 446, 450, 451, 512 [13]; 512 [20]; 514 [65]; 514 [66]; 515 [94]; 516 [120]; 516 [128]; 516 [129] Sigal, I.M 409, 514 [54] Simader, C.G 415, 417, 432, 514 [72] Simon, B 299, 309 [66]; 409, 414, 417–425, 428, 429, 446, 452, 456, 460, 462, 467, 473, 474, 478, 482, 490, 492, 494, 512 [9]; 513 [29]; 516 [112]; 516 [113]; 516 [114]; 517 [130]; 517 [131]; 517 [132]; 517 [133]; 517 [134]; 517 [135]; 517 [136]; 517 [137]; 517 [138] Simon, J 323, 406 [113] Simondon, F 305, 308 [64] Smets, D 17, 55 [83] Smith, H.L 155, 207 [HSW]; 209 [Sm] Smoller, J 199, 206 [CHS] Sobolev, A.V 494, 498–500, 517 [139]; 517 [140]; 517 [141]; 517 [142]; 517 [143]; 517 [144] Sola-Morales, J 314, 344, 402 [24]; 403 [39] Solimini, S 521, 523–529, 532, 557–559, 570, 571, 581, 584, 587, 594, 598 [13]; 598 [14]; 598 [19]; 598 [20]; 599 [25]; 599 [27]; 599 [39]; 599 [40]; 599 [41]; 599 [42]; 599 [43]; 599 [44] Solomyak, M 409, 410, 412, 414, 456, 459, 462–464, 482, 507, 511, 512 [16]; 512 [17]; 512 [18]; 514 [74]; 516 [120]; 516 [121] Solovej, J.P 486, 491, 492, 494, 497, 499–502, 513 [36]; 513 [37]; 513 [38]; 513 [39]; 513 [40]; 515 [80]; 515 [81]; 515 [82]; 515 [83] Solynin, P 9, 17, 53 [34] Spruck, J 21, 54 [54]; 379, 382, 404 [83] Srikanth, P.N 10, 55 [84] Stampacchia, G 319, 321, 406 [114] Steklov, M.W 313, 317, 329, 350, 406 [115] Strang, G 68, 70, 116, 117, 125 [48] Strauss, W.A 9, 55 [85] Struwe, M 3, 33, 35, 55 [86]; 55 [87]; 55 [88]; 523, 527, 530, 539, 571, 581, 584, 594, 598 [14]; 599 [45]; 599 [46] Stubbe, J 472, 499, 500, 512 [19]; 512 [21] Suárez, A 303, 304, 307 [23]; 307 [24]; 307 [25] Sun, J 24, 54 [65] Author Index Suzuki, T 191, 208 [NS] Šverák, V 62, 63, 74, 123, 125 [61]; 125 [62]; 126 [72]; 126 [73] Sweers, G 396, 406 [116] Sychev, M.A 62, 79, 91, 126 [63]; 126 [74]; 126 [75] Szulkin, A 51, 54 [58] Tachizawa, K 465, 469, 517 [145] Tahraoui, R 61, 62, 123 [4]; 123 [5]; 124 [6]; 126 [76]; 126 [77] Taira, K 209 [T] Tamura, H 421, 498, 517 [146]; 517 [147]; 517 [148] Tanaka, K 40, 54 [55] Tang, M 193, 209 [Tang] Tanteri, C 62, 76, 83, 113, 117, 123, 125 [36]; 125 [37] Tehrani, H 35, 53 [28] Terracini, S 27, 31, 53 [44]; 53 [45]; 314, 315, 339, 343, 375, 405 [108]; 405 [109]; 406 [117] Teta, R 421, 512 [3] Thaller, B 425, 478, 480, 492, 515 [86]; 517 [149] Thangavelu, S 493, 517 [150] Thayer, J 389, 406 [118] Thirring, W 454, 468–472, 509, 515 [84]; 515 [85] Thomas, L.E 469, 470, 472, 514 [57] Toledo, J 315, 396, 402 [9]; 402 [12] Tolksdorf, P 321, 347, 356, 376, 406 [119] Torne, O 315, 362–364, 405 [92] Treu, G 62, 126 [78] Tricarico, M 340, 405 [98] Troestler, C 51, 55 [89] Trudinger, N.S 129, 140, 207 [GT]; 240, 307 [33]; 321, 380, 384, 388, 405 [84]; 406 [120] Turner, R.E.L 21, 53 [33] Ubilla, P 40, 54 [71] Uhlenbrock, D.A 431, 474, 514 [53] Umezu, K 314, 342, 406 [121]; 406 [122] Ural’ceva, N 428, 517 [151] van Heerden, F.A 27, 43, 54 [66] Vassiliev, D 409, 516 [123] Vázquez, J.L 347, 352, 354, 376, 397, 398, 404 [78]; 406 [123] Vega, J.M 313, 405 [99] Verbitsky, I.E 433, 436, 438, 439, 442, 444, 515 [95]; 515 [96]; 515 [97]; 517 [152] 607 Véron, L 145, 208 [MV]; 218, 301, 303, 308 [57]; 308 [58]; 309 [67] Verzini, G 31, 53 [45] Vugalter, V 489–491, 513 [41] Waltman, P 207 [HSW] Wang, S.-H 170, 209 [W1]; 209 [W2] Wang, X 425, 514 [51] Wang, Z.-Q 12, 16, 24, 27, 31, 33, 34, 36, 38, 39, 41, 43, 45–47, 49–51, 52 [13]; 52 [14]; 52 [15]; 52 [16]; 53 [38]; 53 [47]; 54 [60]; 54 [61]; 54 [66]; 54 [67]; 54 [68]; 54 [69]; 54 [80]; 55 [90]; 55 [91]; 55 [92]; 55 [93]; 387, 404 [63]; 526, 598 [5] Warzel, S 479, 480, 498, 514 [73]; 516 [111] Wei, J 33, 55 [94] Wei, J.C 18, 54 [75] Weidl, T 456, 469, 471, 472, 514 [56]; 514 [69]; 514 [70]; 517 [153]; 517 [154] Weidmann, J 414, 420, 490, 517 [155] Weinberger, H.F 129, 169, 195, 197, 200, 208 [KW]; 209 [PW]; 301, 308 [63] Weinstein, A 526, 598 [23] Weth, T 17, 24, 27, 31, 33, 34, 38, 42, 52 [12]; 52 [17]; 52 [18]; 52 [19]; 523, 594, 598 [15] Wheeden, R.L 465, 513 [23] Wiebers, H 170, 209 [Wie1]; 209 [Wie2] Willem, M 3, 8, 9, 12, 15, 17, 31, 40, 51, 52 [19]; 52 [20]; 54 [80]; 55 [83]; 55 [89]; 55 [93]; 55 [95]; 55 [96]; 525, 598 [6] Winter, M 33, 55 [94] Wolanski, N 315, 365, 404 [76] Yadava, S.L 344, 361, 368, 369, 402 [3] Yamabe, H 298, 309 [68] Yamada, Y 205, 209 [Y] Yanagida, E 169, 208 [KY] Yang, J 336, 402 [27] Yau, H.-T 486, 491, 499, 515 [87] Yau, S.-T 453, 458, 466, 514 [50]; 514 [75] Yin, H.M 379, 405 [88] Yngvason, J 492, 494, 497, 515 [81]; 515 [82]; 515 [83] Yosida, K 84, 126 [79] Zagatti, S 62, 114, 124 [19]; 126 [80] Zhu, M 314, 343, 346, 369, 405 [93]; 405 [94]; 406 [124] Zou, W 16, 55 [82]; 315, 388, 390, 405 [111]; 406 [112]; 406 [125] This page intentionally left blank Subject Index CLR estimate, 412 compact operator, 503 competition system, 193 concentration–compactness, 521, 525 – method, 330, 340 constant electric field, 429 continuous spectrum, 504 convergence quasieverywhere, 434 convex – envelope, 73 – – , definition, 67 – hull, 74, 81 – – , definition, 75 – integration, 123 Coulomb gauge, 415 coupling constant, 412 Crandall and Rabinowitz transversality condition, 237 critical – dimension, 368 – exponent, 315, 322, 340, 343, 367 critical growth, 521, 522, 524, 526, 527, 529, 539, 548, 587 crowding effects, 216 Cwikel – inequality, 455 – – , operator-valued case, 463 A a priori bounds of Keller and Osserman, 218, 235 adjoint operator, 504 Aharonov–Bohm – effect, 415 – magnetic potential, 421 Aharonov–Casher theorem, 487 Alibert–Dacorogna–Marcellini example, 65 anticommutation relation, 416 antimaximum principle, 395 approximation property, 81, 115 – , definition, 80 attracting property, 223 B Baire category theorem, 79, 84 band subspace, 493 Bessel capacity, 434 bifurcation – from a simple eigenvalue, 130 – from infinity, 131 Birman–Schwinger – operator, 440, 441, 453, 511 – principle, 446, 510 blow-up technique, 382 Bohr–Sommerfeld quantization condition, 411 Bolza example, 60, 106 boundary blow-up, 133 D Dacorogna formula, 68 decay estimates, 541 deformation lemma, 14 diamagnetic – inequality, 474 – – , strong, 443 – – , weak, 424 – monotonicity, 477 diamagneticity, 424 differential – expression, 414 – operation, 414 distribution function, 506 C Carathéodory theorem, 68, 102, 103, 106 classical – constant, 411 – Hamiltonian, 409 – logistic equation, 234, 235 – solution, 219, 223 closable – form, 420 – – , relatively, 422 – operator, 503 closed operator, 503 closure, 417 609 610 domain of the operator, 503 domination, 443 double natural constraint, 570, 581, 594, 596 E eigenvalue cluster, 498 eigenvalues, 411 electric potential, 416 elliptic system, 378 embedding theorem, 419 equivalent constants, 437 essential spectrum, 505 essentially self-adjoint operator, 417, 504 evolution equation, 426 exchange stability principle, 239 explosive solution, 216, 218 F Feynman–Kac formula, 456 finite propagation speed property, 427 Fredholm analytic pencil, 236 free boundary, 395 Friedrichs extension, 421 Fuˇcík spectrum, 400, 401 function of first class, 84 G gauge transformation, 414 Gelfand equation, perturbed, 169 genus, 325 geodesically complete manifold, 427 Glazman lemma, 494, 508 gradient elliptic systems, 390 group action, 11 growth rate – , boundary, 278, 279 – , intrinsic, 216 H Hamiltonian elliptic systems, 385 harmonic oscillator, 419 Harnack inequality, 300 Hausdorff measure, 434 heterogeneous competition system, 147 Holmgren principle, 427 I indefinite superlinear parabolic problems, 305 infinitesimal form-boundedness, 433 J Jensen inequality, 73, 81, 103 Subject Index K K-attractive, 26 Kato inequality, 431 Kato–Rellich theorem, 418 KLMN theorem, 422 L lamination convex hull, definition, 75 Landau – bands, 493 – levels, 487, 492 Landesman–Lazer condition, 399, 400 Laplace–Beltrami operator, 414 Laplacian, 414 – , magnetic, 414 large – solution, 216, 218, 220, 278 – – , continuity, 296 – – , strong monotonicity, 295 layer solution, 344 Lieb–Thirring inequality, 413 – for operator-valued potentials, 471 – for Pauli operators, 492, 498 Lieb–Thirring inequality, 468, 494 linear operator, 503 localization method, 303 M magnetic – bottle, 450 – field, 414 – gradient, 414 – potential, 414 Malthus law, 216, 217 maximal – operator, 418 – solution, 219, 226 metasolution, 216, 220, 223 method of moving spheres, 343 min–max, 521, 523, 570, 578, 580, 581, 583, 584, 588–590, 597 minimal – operator, 419 – solution, 219, 226 Molchanov functional, 449 mountain pass theorem, 333 multibump solution, 45 multiplicity, 521, 523, 524, 526, 527, 536, 539, 541, 570, 593, 594 – , exact, 169 N Nehari manifold, 5, 13 Neumann–Schatten class, 508 Subject Index nodal – domain, 27 – Nehari set, 22 – properties, 193 nod(u), 27 nonlinear boundary conditions, 313, 314, 317, 319, 322, 323, 378 nonvariational systems, 378 null Lagrangian, 63 P p-capacity, 367 Palais–Smale condition, 323, 324, 328, 330, 331, 334, 335, 394 paramagnetic monotonicity, 477 patterned solution, 137 Pauli – matrices, 415 – operator, 413, 415 phase-space volume, 452, 467 Pohozaev – formula, 546, 550 – identity, 338, 361, 522, 523, 541, 544, 546 – inequality, 527, 541, 544 Poincaré gauge, 479 point spectrum, 504 polyconvex – envelope, definition, 67 – function, 64, 66, 95, 99, 100 – – , definition, 62 – hull, definition, 75 population behavior, 217 porous medium equation, 304 positively – dominated, 473 – invariant, 24 – – , strictly, 24 – preserving, 443, 460 – – semigroups, 460 potential wells, 121, 122 predator–prey system, 200 Q quadratic form, 419 – , closed, 419 quasiaffine function, 63–65, 68, 87–89, 110, 112–114, 116–118, 120 – , definition, 62 quasiconvex – envelope, 69, 71, 86, 115, 117 – – , definition, 67 – function, 64, 72, 78, 79, 86, 92 – – , definition, 62 – hull, definition, 75 611 R Rabinowitz – theorem, 343 – – , global bifurcation, 130 rank one convex – envelope, definition, 67 – function, definition, 62 – hull, 76 – – , definition, 75 relatively – bounded operator, 418 – – , infinitesimally, 418 – compact forms, 436 relaxation – property, 77, 78, 80, 81, 87–91, 112, 113, 115, 117 – – , definition, 77 – theorem, 60, 70, 86 resolvent – operator, 236 – set, 504 resonance, 399, 400 Riesz capacity, 434 S s-numbers, 507 Saint Venant–Kirchhoff energy, 69 Schatten ideals, 473 Schrödinger – equation, 411 – operator, 409 – – , magnetic, 431, 450 Schwarz – symmetric, foliated, 17, 31 – symmetrization, – – , foliated, 17 self-adjoint – extension, 417 – operator, 504 separability, 428 separation theorem, 64, 68 sesquilinear form, 420 simple eigenvalue, 131 singular values, 69, 76, 77, 82, 110, 111, 115 – , definition, 65 Sobolev – constant, 357 – embedding, 320, 358, 361, 534, 537, 539, 565 – inequality, 423 – space, 316, 317, 417 – trace, 317, 340 – – constant, 316, 317, 355, 356, 365, 367, 371, 375, 377 612 – – embedding, 318, 331, 358, 360, 367 – – inequality, 322, 351 – – theorem, 313, 316, 326, 333, 361 spatial degeneracy, 131 spectral – measure, 505 – theorem, 505 spectrum, 504 spherical symmetrization, 364, 366 spherically symmetric, 364, 366 stability – analysis, 154 – of matter, 468 stable pattern, 163 Stark potential, 419 Steklov eigenvalues, 313, 317, 329, 350 strictly convex function in at least N directions, 95, 97, 101 – , definition, 94 strictly quasiconvex function, 94, 97, 99 – , definition, 92 strong maximum principle, 226 strongly increasing, 256 structural stability, 240 Stummel class, 428 subcritical – exponent, 322, 342 – growth, 521, 524 subsolution, 226 Subject Index supersolution, 226 – element, 254 – positive, strict, 226 Šverák example, 63 symmetric operator, 504 symmetry and symmetry breaking, 362 T trace class operator, 507 trace-property, 371 Trotter–Kato–Masuda formula, 474 turning point theorem, 131 U unbounded domains, 533 V Verhulst law, 216, 217 W weak lower semicontinuity, definition, 64 Wiener – capacity, 433 – integral, 456 Y Yamabe problem, 344 Z zero modes, 413, 486 – density, 491 ... 2. 3 Upper bounds on the number of nodal domains HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume Edited by M Chipot and P Quittner © 20 05... (f0 ) with p < 2 , (f1 ) and (f3 ) Moreover, suppose that − + a is positive Then there exists a minimizer of Φ on N+ and, hence, a positive solution of (1.1 2) P ROOF Let (un ) ⊂ N+ be a minimizing... (1.1 6) where the domain Ω has some symmetry We shall prove, by the method of moving planes, that, under some assumptions, all the solutions of (1.1 6) inherit the symmetry of Ω The method of moving