PARTIAL DIFFERENTIAL EQUATIONS
1 THE DIRICHLET PROBLEM FOR SUPERLINEAR ELLIPTIC EQUATIONS 1 2 NONCONVEX PROBLEMS OF THE CALCULUS OF VARIATIONS AND DIFFERENTIAL INCLUSIONS 55 3 BIFURCATION AND RELATED TOPICS IN ELLIPTIC PROBLEMS 124 4 METASOLUTIONS MALTHUS VERSUS VERHULST IN POPULATION DYNAMICS A DREAM OF VOLTERRA 207 5 ELLIPTIC PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS AND THE SOBOLEV TRACE THEOREM 306 6 SCHR DINGER OPERATORS WITH SINGULAR POTENTIALS 401 7 MULTIPLICITY TECHNIQUES FOR PROBLEMS WITHOUT COMPACTNESS 512 CONTENTS CHAPTER 1 THE DIRICHLET PROBLEM FOR SUPERLINEAR ELLIPTIC EQUATIONS 1 0 INTRODUCTION BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS HAVE BEEN A MAJOR FOCUS OF RESEARCH IN NONLINEAR ANALYSIS FOR DECADES IN THIS SURVEY WE DISCUSS SEMILINEAR EQUATIONS LIKE U A X U F X U X U 0 ON 0 1 WHERE IS A DOMAIN IN RN N 2 AND F R R IS SUPERLINEAR THAT IS F X T T AS T THE MODEL NONLINEARITY IS THE HOMOGENEOUS FUNCTION F X T T P 2T WITH P 2 0 2 THE CONTINUATION METHOD OR OTHER CLASSICAL METHODS BASED ON THE LERAY SCHAUDER DE GREE DO NOT APPLY EASILY TO 0 1 BECAUSE THERE ARE NO A PRIORI BOUNDS FOR THE SOLUTIONS THIS IS INHERENT TO SUPERLINEAR NONLINEARITIES IN FACT FOR THE MODEL NONLINEARITY 0 2 WITH P 2 P 2N N 2 IF N 3 A L AND BOUNDED THERE EXIST IN NITELY MANY SOLUTIONS WHICH ARE UNBOUNDED IN THE H 1 NORM ON THE OTHER HAND IF RN IS STAR SHAPED A X CONST 0 F IS AS IN 0 2 WITH P 2N N 2 N 3 THEN 0 1 HAS NO SOLUTION EXCEPT THE TRIVIAL ONE U 0 HERE WE KNOW A POSTERIORI THAT THE SOLUTIONS ARE BOUNDED BUT THE LERAY SCHAUDER METHODS DO NOT APPLY DUE TO A LACK OF COMPACTNESS AFTER SOME INITIAL WORK DURING THE 1960S AND EARLY 1970S AMBROSETTI AND RABINOWITZ ESTABLISHED IN THE SEMINAL PAPER 5 SEVERAL VARIATIONAL METHODS TO OBTAIN SOLUTIONS OF 0 1 ON BOUNDED DOMAINS MOST NOTABLY THE MOUNTAIN PASS THEOREM AND VARIATIONS THEREOF THESE METHODS HAVE BEEN RE NED AND EXTENDED TO DEAL FOR INSTANCE WITH UNBOUNDED DO MAINS OR THE CRITICAL EXPONENT CASE F X T B X T 4 N 2 T WHICH IS CLOSELY RELATED TO THE YAMABE PROBLEM FROM DIFFERENTIAL GEOMETRY WITH THESE METHODS MORE COMPLICATED PARTIAL DIFFERENTIAL EQUATIONS WITH VARIATIONAL STRUCTURE CAN NOW BE INVESTIGATED MOREOVER QUALITATIVE PROPERTIES OF THE SOLUTIONS HAVE BEEN DISCOVERED IN RECENT YEARS IN PARTICULAR ON THE NODAL STRUCTURE AND THE SYMMETRY OF THE SOLUTIONS THE GOAL OF THIS CHAPTER IS TO PRESENT SOME BASIC IDEAS IN A SIMPLE SETTING AND TO SURVEY SELECTED RESULTS ON THE DIRICHLET PROBLEM 0 1 WITH SUPERLINEAR NONLINEARITY THE CHAP TER CONSISTS OF THREE SECTIONS IN SECTION 1 WE DEAL WITH POSITIVE SOLUTIONS OF 0 1 IN SECTION 2 WITH SIGN CHANGING SOLUTIONS ON BOUNDED DOMAINS AND IN SECTION 3 WE TREAT THE UNBOUNDED DOMAIN RN NO EFFORT IS BEING MADE TO BE AS GENERAL AS POSSIBLE NEITHER DID WE TRY TO WRITE A COMPREHENSIVE SURVEY ON 0 1 FOR EXAMPLE WE DO NOT PRESENT RESULTS ON THE BIFURCATION OF SOLUTIONS NOR FOR THE P LAPLACE OPERATOR NOR DO WE TREAT SINGULARLY PERTURBED EQUATIONS IN DETAIL THESE TOPICS REQUIRE SEPARATE SURVEYS FOR TUNATELY THERE ARE A NUMBER OF WELL WRITTEN MONOGRAPHS ABOUT 0 1 WHERE THE READER CAN ND ADDITIONAL INFORMATION AND FURTHER REFERENCES WE WOULD LIKE TO MENTION THE BOOKS BY RABINOWITZ 79 STRUWE 87 CHANG 41 GHOUSSOUB 52 KAVIAN 56 SCHECHTER 81 WILLEM 95 CHABROWSKI 39 40 KIELH FER 57 WE CONCENTRATE ON TOPICS NOT BEING COV ERED IN THESE MONOGRAPHS ALTHOUGH A CERTAIN OVERLAP CANNOT BE AVOIDED FOR NATURAL REASONS OF COURSE THE CHOICE OF TOPICS IS ALSO IN UENCED BY OUR OWN RESEARCH INTERESTS PARTIAL DIFFERENTIAL EQUATIONS MA 3132 LECTURE NOTES 2 0 1 CONDITIONS ON THE NONLINEARITY FOR THE CONVENIENCE OF THE READER WE LIST HERE CONDITIONS ON F WHICH WE USE AT VARIOUS PLACES IN THE CHAPTER THE CRITICAL EXPONENT IS DE NED BY 2 IF N 2 2N N 2 IF N 3 GIVEN F R R WE LET F R R F X T T 0 F X S DS BE THE PRIMITIVE OF F F0 F R R IS A CARATH ODORY FUNCTION WITH F X T O T AS T 0 THERE EXIST C 0 AND P 2 SUCH THAT F X T C T T P 1 FOR ALL X T R F 0 F R R IS DIFFERENTIABLE IN T R AND THE DERIVATIVE FT IS A CARATH ODORY FUNCTION WITH FT X 0 0 THERE EXIST C 0 AND P 2 SUCH THAT FT X T C 1 T P 2 FOR ALL X T R F1 FOR EVERY X THE FUNCTION R 0 R T F X T T IS STRICTLY INCREASING F 1 FT X T F X T T FOR EVERY X AND EVERY T 0 F2 THERE EXIST R 0 AND 2 SUCH THAT 0 F X T F X T T FOR ALL X T R IF RN IT IS REQUIRED THAT R 0 F3 LIM T F X T T2 UNIFORMLY IN X F4 THERE EXISTS M 0 SO THAT T F X T MT IS STRICTLY INCREASING FOR ALL X F5 INFT 0 F X T T 1 POSITIVE SOLUTIONS 1 1 EXISTENCE OF POSITIVE SOLUTIONS WE CONSIDER RST THE PROBLEM U A X U UP 1 X U 0 X U 0 ON 1 1 WHERE IS A SMOOTH DOMAIN IN RN A L AND 2 P 2 IN ORDER TO OBTAIN A SOLUTION OF 1 1 WE ASSUME THAT A IS POSITIVE I E THERE EXISTS C 0 SUCH THAT FOR EVERY U H 1 0 U 2 A X U2 DX C U2 DX 1 2 OUR MAIN TOOL IS THE RELLICH COMPACTNESS THEOREM THEOREM 1 1 LET BE BOUNDED AND LET 1 P 2 THEN THE INJECTION H 1 0 LP IS COMPACT PARTIAL DIFFERENTIAL EQUATIONS MA 3132 LECTURE NOTES 3 THEOREM 1 2 ASSUME THAT IS BOUNDED 2 P 2 AND 1 2 IS SATIS ED THEN THERE IS A SOLUTION OF 1 1 PROOF BY THEOREM 1 1 AND LOWER SEMICONTINUITY IT IS EASY TO VERIFY THAT INF V H1 0 V LP 1 V 2 A X V2 DX 1 3 IS ACHIEVED BY SOME V AFTER REPLACING V BY V WE MAY ASSUME THAT V 0 IT FOLLOWS FROM THE LAGRANGE MULTIPLIER RULE THAT V A X V VP 1 A SOLUTION OF 1 1 IS THEN GIVEN BY U 1 P 2 V INDEED U 0 ON BY THE STRONG MAXIMUM PRINCIPLE REMARK 1 3 THERE ARE OTHER WAYS TO PROVE THEOREM 1 2 INSTEAD OF MINIMIZING AS IN 1 3 ONE CAN MINIMIZE THE FUNCTIONAL U 1 2 U 2 A X U2 DX 1 P U P DX ON THE NEHARI MANIFOLD N U H 1 0 U 0 U U 0 U H 1 0 U 0 U 2 A X U2 DX U P DX A CRITICAL POINT OF THE CONSTRAINED FUNCTIONAL N IS A CRITICAL POINT OF HENCE A SOLUTION OF 1 1 THE MAP V H 1 0 V LP 1 N V V 2 A X V2 DX 1 P 2 V IS A DIFFEOMORPHISM WITH INVERSE N V H 1 0 V LP 1 U U U LP IT MAPS THE MINIMIZER V OF 1 3 TO THE MINIMIZER U OF ON N THE SOLUTION CAN ALSO BE OBTAINED VIA THE MOUNTAIN PASS THEOREM FROM 5 IN FACT U INF U 0MAX T 0 TU INF MAX T 0 1 T PARTIAL DIFFERENTIAL EQUATIONS MA 3132 LECTURE NOTES 4