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Marino Badiale  Enrico Serra

Semilinear Elliptic Equations for Beginners Existence Results via the Variational Approach

Marino Badiale Dipartimento di Matematica Università di Torino Torino, Italy [email protected]

Enrico Serra Dipartimento di Matematica Università di Torino Torino, Italy [email protected]

Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, Università degli Studi di Milano, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus J. Macintyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, CNRS, École Polytechnique, Palaiseau, France Endre Süli, University of Oxford, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA

ISBN 978-0-85729-226-1 e-ISBN 978-0-85729-227-8 DOI 10.1007/978-0-85729-227-8 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Mathematics Subject Classification (2000): 35-01, 35A15, 35B38, 35J15, 35J20, 35J25, 35J60, 35J61, 35J92 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is an introduction to the study of a wide class of partial differential equations, namely semilinear elliptic problems, through the application of variational methods and critical point theory. Semilinear elliptic equations arise in a variety of contexts in geometry, physics, mechanics, engineering and, more recently, in life sciences. None of these fields can be investigated without taking into account nonlinear phenomena, and variational methods, as a branch or an evolution of the Calculus of Variations, are almost entirely concerned with nonlinearity. The theory has attained a spectacular success in the last thirty-or-so years, reachin