a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

infinitely many solutions for symmetric and non symmetric elliptic systems we study an elliptic system equivalent to a fourth order elliptic equation by using variational and perturbative methods we prove the existence of infinitely many solutions both in the symmetric and in the non symmetric case

a non oscillation theorem for the emden fowler equation ground states for semilinear elliptic equations with critical exponents abstract let be a ball centered at the origin in r n n 2 its boundary and 0 0 a given function this article is concerned with radial solutions of the boundary value problem u x u p 0 u 0 in u 0 on 1 radial solutions of 1 are functions u of the variable r x that satisfy the ordinary differential equation u n 1 r u r u p 0 2 it is shown that if p p where p is the critical sobolev exponent p n 2 n 2 then 2 is non oscillatory at the origin if there exists a 0 such that the function r r log 1 r is non decreasing near the origin the result is a corollary of a similar result for the emden fowler equation x t 2 1 g t x 1 2 0 3 where g is a given positive valued function and a positive constant the relation between the solutions of 2 and 3 is established via the transformations x t u r g t r where t n 2 r n 2 it is shown that 3 is nonoscillatory at infinity if there exists a 0 such that the function t g t log t is non increasing for all sufficiently large t the proof of this result constitutes the core of the article the result is of independent interest in the theory of the emden fowler equation and improves earlier oscillation results of nehari and chiou

elliptic equations with nearly critical growth n f 2 n 2 it has no solution for any star shaped domain 13 recently considerable interest has grown around problems like i in which the right hand side up is replaced by a perturbation f u of the pure power such as f u au up where 1 e iw and 0

variational identities and applications to differential systems j serrin 1 introduction in the work of ponos p and in recent work of pucci serrin ps variational identities are discussed which are useful for solving various ques tions about elliptic differential equations and particular elliptic systems here we obtain a number of further applications of these identities to systems it was noted in ps that these identities are closely related to noether s theorem for variational problems by working out the divergence term arising in noether s theorem and making some simplifications one obtains a general identity for variational problems if a variational integral has a symmetry group g with the following infinitesimal generator

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

some applications of the pohozaev identity we apply the pohozaev identity to show the nonexistence of nontrivial solutions to a semilinear equation of the form h e u f x u u where e is a real constant lying on the essential spectrum of h the self adjoint operators h under consideration are the schrodinger operator with coulomb type potentials the stark like hamiltonian and the semirelativistic hamiltonian

a note on superlinear hamiltonian elliptic systems this paper is concerned with the superlinear periodic elliptic systems of hamiltonian type in the whole space the existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

rellich type identities for eigenvalue problems and application to the pompeiu problem for classical neumann eigenvalue buckling eigenvalue and clamped plate eigenvalue we give the corresponding rellich type identities as an application of these results then we obtain a new necessary and sufficient condition for a domain without the pompeiu property

computing modular polynomials in quasi linear time we analyse and compare the complexity of several algorithms for computing modular polynomials we show that an algorithm relying on floating point evaluation of modular functions and on interpolation which has received little attention in the literature has a complexity that is essentially up to logarithmic factors linear in the size of the computed polynomials in particular it obtains the classical modular polynomials phi ell of prime level ell in time o ell 3 log 4 ell log log ell besides treating modular polynomials for gamma 0 ell which are an important ingredient in many algorithms dealing with isogenies of elliptic curves the algorithm is easily adapted to more general situations composite levels are handled just as easily as prime levels as well as polynomials between a modular function and its transform of prime level such as the schlafli polynomials and their generalisations our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors

on the classification of standing wave solutions to a coupled schrodinger system in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball and entire space as the parameters within certain regions in addition a complete structure of different types of solutions for the radial case is also provided

on the uniqueness and structure of solutions to a coupled elliptic system abstract in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball in addition the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved

uniqueness results for semilinear polyharmonic boundary value problems on conformally contractible domains i abstract we study polyharmonic boundary value problems mu f u m n with dirichlet boundary conditions on bounded and unbounded conformally contractible domains in r n such domains can be contracted to a point bounded case or to infinity unbounded case by one parameter groups of conformal maps the class of star shaped domain is a subclass the problem has variational structure this allows us to derive a sufficient condition for uniqueness by studying the interaction of one parameter transformation groups with the underlying functional l if the transformation group strictly reduces the values of l then uniqueness of the critical point of l follows the proof is inspired by e noether s theorem on symmetries and conservation laws applications of the uniqueness principle are given in part ii of this paper

multiple solutions for elliptic systems with nonlinearities of arbitrary growth abstract we prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth moreover if the symmetry of the problem is broken by a small enough perturbation term we find at least three solutions the proofs utilise a variational setting given by de figueiredo and ruf in order to prove an existence s result and the algebraic approach based on the pohozaev s fibering method

asymptotically linear schrodinger equations with sign changing potential abstract in this paper we show that the following system 0 1 u v x u g x v v v x v f x u x r n possesses at least one non trivial solution pair u v where f x t g x t are continuous functions on r n r and asymptotically linear as t and v x is allowed to be sign changing

asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth ii the nonradial case abstract we consider positive solutions ue of a family of elliptic boundary value problems ie as e 0 they approach in some sense the problem i0 which has a nonlinearity with critical or supercritical growth it is shown how the solution ue concentrates at a point how this happens asymptotically and where in the domain it happens

on a fourth order elliptic equation involving the critical sobolev exponent the effect of the graph topology abstract in this paper we consider the following nonlinear equation 2 u k u n 4 n 4 e u with u 0 in and u u 0 on where is a smooth bounded domain in r n n 8 k is a smooth positive function in and e is a small positive parameter using the effect of the graph topology of the function k we prove existence multiplicity and concentration of solutions we prove also a nonexistence result

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

k theoretic amenability for sl2 qp and the action on the associated tree abstract it is proved that if a locally compact group g acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable then g is k amenable in particular the canonical map from the full c algebra onto the reduced c algebra of g induces isomorphisms in k theory the main corollary of our result is that sl 2 q p and some other groups over local fields are k amenable

multiplicity of positive solutions for p laplace equation with superlinear type nonlinearity abstract in this work we consider the problem p r p u u p u q u 0 in b r 0 u 0 on b r 0 where p 2 p np n p and b r 0 is the open ball of radius r centered about the origin in r n n 3 under the condition that p 1 q p 2 n p 1 we show that there exists r 0 such that p r admits atleast two solutions for all r r and no solution if r this extends the results of nonlinear anal tma 10 8 1986 755 to the case of p laplace equation

solutions of elliptic equations with critical sobolev exponent in dimension three theorem 1 for every a 72 4 there exists a solution of i the previously known results concerning this problem are the following 1 if 1 5 0 i has no solution this follows from pohozaev s identity see lo and from the principle of unique continuation when a 0 we will suppose in the following that i 0 2 every positive solution of i is radial see gidas et al 9 3 let a be the first eigenvalue of a in b with dirichlet boundary condition and p1 the associated eigenfunction then for every a 2 i i has no positive solution this result follows from multiplying i by cp 4 for every a e n2 4 rr there exists a positive solution of i see brezis and nirenberg 5 5 for i 5 n2 4 i has no radial solution see 5 6 for every k e n if 1 e zk with zk k t n k2n2 i has a radial solution with k 1 zeros see atkinson and peletier 2 31 7 the analogue of theorem 1 in dimension n 2 4 holds for general domains see capozzi et al 6 and ambrosetti and struwe l our method is very different and we do not know whether theorem 1 is valid for general domains in ir3

on a class of semilinear elliptic systems abstract this paper deals with the existence of positive solutions to a dirichlet problem for the superlinear elliptic system u f v v g u on a ball where is a positive parameter and f 0 g 0 may be negative

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

strongly indefinite systems with critical sobolev exponents we consider an elliptic system of hamiltonian type on a bounded domain in the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms our proof is based on an adaptation of the dual variational method as applied before to the scalar case introduction existence and non existence of solutions of semilinear elliptic srstems has been a subject of active research recently see for example ff l3 ps such systems are called variational if solutions can be viewed as critical points of an associated functional defined on a suitable function space restricting our attention to systems with two unknowns we distinguish two classes of variational elliptic systems a potential systems b hamiltonian systems potential systems are of the form a llu u v ztl2v u v where l1 and l2 are second order selfadjoint elliptic operators formally these are the euler lagrange equations of the functional f u v l1u u i l2v v th u v here is the l2 inner product for functions defined on the underlying domain and tt u v s h u x v x dx the functional f is often called the lagrangian its critical points correspond to weak soltbtions of a the appropr ate choice of the function spaces for u and v follows by requiring that the quadratic part of f be well defined this naturally leads to sobolev spaces with square irltegrable first derivatives since also a must be well defined by virtue of the embedding theorems some growth restrictions on this latter function are required an extra difficulty appears if we consider system a with a plus sign in the second equation for then the corresponding functiorl f has a strongly indefinite quadratic part for systems of superlinear type and with sllbcritical growth the mountain pass theorem ar or in the strongly indefinite case the benci rabinowitz theorem received by the editors june 5 1996 1991 mathematics subject classification primary 35j50 35j55 35j65

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

groups acting properly on bolic spaces and the novikov conjecture we introduce a class of metric spaces which we call bolic they include hyperbolic spaces simply connected complete manifolds of nonpositive curvature euclidean buildings etc we prove the novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a bolic weakly geodesic metric space of bounded geometry

amenability and exactness for dynamical systems and their c algebras we study the relations between amenability resp amenability at infinity of c dynamical systems and equality or nuclearity resp exactness of the corresponding crossed products

existence of solutions for a class of noncooperative elliptic systems we establish the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of superlinear growth moreover if there is resonance we also find at least a solution all results are obtained by the minimax methods in critical point theory

existence and uniqueness of positive radial solutions for the lane emden system in this article we consider component wise positive radial solutions of a weakly coupled system of elliptic equations in a ball with homogeneous nonlinearities the existence is well known in general we give a result for the remaining cases the uniqueness is less studied we complement the known results

on a new class of elliptic systems with nonlinearities of arbitrary growth abstract we guarantee the existence of infinitely many different pairs of solutions to the system u v p in v f u in u v 0 on where 0 p 2 n 2 is a bounded domain in r n and the continuous nonlinear term f has an unusual oscillatory behavior the sequence of solutions tends to zero resp infinity with respect to certain norms and the nonlinear term f may enjoy an arbitrary growth at infinity resp at zero whenever f oscillates near zero resp at infinity our results provide the first applications of ricceri s variational principle in the theory of coupled elliptic systems

e theory and kk theory for groups which act properly and isometrically on hilbert space a good deal of research in c algebra k theory in recent years has been devoted to the baum connes conjecture 3 which proposes a formula for the k theory of group c algebras that blends group homology with the representation theory of compact subgroups the conjecture has brought c algebra theory into close contact with manifold theory through its obvious similarity to the borel conjecture of surgery theory 9 31 and its links with the theory of positive scalar curvature 27 in addition there are points of contact with harmonic analysis particularly the tempered representation theory of semisimple groups although the proper relation between the baum connes conjecture and representation theory is not well understood the conjecture is most easily formulated for groups which are discrete and torsion free for such a group g there is a natural homomorphism

asymptotically linear elliptic systems in rn abstract in this paper we show that semilinear elliptic systems of the form u u v v g v v v u u f u x r n possess at least one nontrivial solution pair u v h 1 r n h 1 r n where and are positive numbers f t and g t are continuous functions on r and asymptotically linear as t

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method