the quasiclassical limit of the modified kp hierarchy in the quasiclassical limit the mkp hierarchy and its lax representation turn into a modified benney hierarchy and its poisson representation a miura map is constructed and shown to be canonical from the modified benney hierarchy into the unmodified one the modified hierarchy is given both hydrodynamical and kinetic representations and the miura map is given a kinetic form explicit combinatorial formulae are proved for the infinite number of conserved densities of the modified benney hierarchy

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

solvable vector nonlinear riemann problems exact implicit solutions of dispersionless pdes and wave breaking we have recently solved the inverse spectral problem for integrable partial differential equations pdes in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter the associated inverse problem in particular can be formulated as a nonlinear riemann hilbert nrh problem on a given contour of the complex plane the most distinguished examples of integrable pdes of this type like the dispersionless kadomtsev petviashivili dkp the heavenly and the two dimensional dispersionless toda equations are real pdes associated with hamiltonian vector fields the corresponding nrh data satisfy suitable reality and symplectic constraints in this paper generalizing the examples of solvable nrh problems illustrated in manakov and santini 2009 j phys a math theor 42 095203 2008 j phys a math theor 41 055204 2009 j phys a math theor 42 404013 we present a general procedure to construct solvable nrh problems for integrable real pdes associated with hamiltonian vector fields allowing one to construct exact implicit solutions of such pdes parametrized by an arbitrary number of real functions of a single variable then we illustrate this theory on few distinguished examples for the dkp and heavenly equations for the dkp case we characterize a class of similarity solutions of solutions constant on their parabolic wave front and breaking simultaneously on it of localized solutions whose breaking point travels with constant speed along the wave front and of localized solutions breaking in a point of the x y plane for the heavenly equation we characterize two classes of symmetry reductions

on twistor solutions of the dkp equation the factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered a method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dkp equation

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

the homological torsion of psl_2 of the imaginary quadratic integers denote by q sqrt m with m a square free positive integer an imaginary quadratic number field and by a its ring of integers the bianchi groups are the groups sl 2 a we reveal a correspondence between the homological torsion of the bianchi groups and new geometric invariants which are effectively computable thanks to their action on hyperbolic space we expose a novel technique the torsion subcomplex reduction to obtain these invariants we use it to explicitly compute the integral group homology of the bianchi groups furthermore this correspondence facilitates the computation of the equivariant k homology of the bianchi groups by the baum connes conjecture which is verified by the bianchi groups we obtain the k theory of their reduced c algebras in terms of isomorphic images of their equivariant k homology

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

on integration of multidimensional generalizations of classical c and s integrable nonlinear partial differential equations we develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical c and s integrable partial differential equations pdes generalizations of 1 1 dimensional c integrable and 2 1 dimensional s integrable n wave equations are derived among examples examples of multidimensional second order pdes are represented as well

decaying solution for singular p laplacian equation abstract using a variational approach we study a class of singular quasilinear elliptic equation in r n under suitable assumptions on the measurable coefficient f x and the locally lipschitz function g u we show the existence of the radial decaying solution for the problem

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

proving modularity for a given elliptic curve over an imaginary quadratic field we present an algorithm to determine if the l series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree by the work of harris soudry taylor taylor and berger harcos cf hst93 tay94 and bh we can associate to an automorphic representation a family of compatible p adic representations our algorithm is based on faltings serre s method to prove that p adic galois representations are isomorphic modularity for rational elliptic curves was one of the biggest achievements of last century little is known for general number fields in the case of totally real number fields some techniques do apply but the result is far from being proven the case of not totally real fields is more intractable to taylor wiles machinery in this paper we present an algorithm to determine if the l series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree or not the algorithm is based on faltings serre s method to prove isomorphism of p adic galois representation by the work of harris soudry taylor taylor and berger harcos cf hst93 tay94 and bh we can associate to an automorphic representation a family of compatible p adic representations and an elliptic curve has such a family of representations as well in the natural way the paper is organized as follows on the first section we present the algorithms which depend on the residual representations on the second section we review the results of p adic representations attached to automorphic forms on imaginary quadratic fields on the third section we explain falting serre s method on galois representations on the fourth section we prove that the algorithm gives the right answer at last we show some examples and some gp code writen for the examples 2 algorithm

stickelberger elements and modular parametrizations of elliptic curves in the present paper we shall give evidence to support the claim conjecture i below and 1 3 that every elliptic curve a o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of a in particular we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of stickelberger elements ad p adic l functions attached to a in addition if conjecture i is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the q isogeny class of a distinguished by modular considerations for most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of x i n definition 1 1 however to justify our view of these parametrizations as being canonical we begin here with a more intrinsic definition recall that shimura 19 chap 6 see w 1 of this paper has defined a compatible system of canonical models of modular curves xs ses where 5 p is a certain collection of open subgroups of the group gl 2 az over the finite adeles a i of q we define the adelic upper half plane to be the pro variety ll m xs and give the q structure induced by the s field of modular functions whose q expansions at the 0 cusp have coefficients in q a modular parametrization of a is a q morphism a which sends

binary hermitian forms over a cyclotomic field abstract let be a primitive fifth root of unity and let f be the cyclotomic field f q let o f be the ring of integers we compute the voronoi polyhedron of binary hermitian forms over f and classify gl 2 o conjugacy classes of perfect forms the combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group gl 2 o and hecke eigenforms

the inverse spectral transform for the dunajski hierarchy and some of its reductions i cauchy problem and longtime behavior of solutions in this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations pdes arising from the commutation condition of pairs of one parameter families of vector fields recently developed by s v manakov and one of the authors to one distinguished class of equations the so called dunajski hierarchy we concentrate for concreteness i on the system of pdes characterizing a general anti self dual conformal structure in neutral signature ii on its first commuting flow and iii on some of their basic and novel reductions we formally solve their cauchy problem and we use it to construct the longtime behavior of solutions showing in particular that unlike the case of soliton pdes different dispersionless pdes belonging to the same hierarchy of commuting flows evolve in time in very different ways exhibiting either a smooth dynamics or a gradient catastrophe at finite time

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

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

solutions of the dispersionless toda hierarchy constrained by string equations solutions of the riemann hilbert problem implementing the twistorial structure of the dispersionless toda dtoda hierarchy are obtained two types of string equations are considered which characterize solutions arising in hodograph sectors and integrable structures of two dimensional quantum gravity and laplacian growth problems

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

minkowski s conjecture well rounded lattices and topological dimension let a c sln r be the diagonal subgroup and identify sln r sln z with the space of unimodular lattices in rn in this paper we show that the closure of any bounded orbit a lcsln r sln z meets the set of well rounded lattices this assertion implies minkowski s conjecture for n 6 and yields bounds for the density of algebraic integers in totally real sextic fields the proof is based on the theory of topological dimension as reflected in the combinatorics of open covers of rn and tn

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

the initial value problem for the dkp hierarchy the solution corresponding to each initial condition for the dispersionless kp hierarchy can be found from the integration of a hamilton jacobi equation by means of a transformation of coordinates the solution is explicitly determined in a parametric form as well as a power series in a deformation parameter in addition a twistor formulation of the solution is given

on the continuous benney equations abstract the continuous benney equations and its riemann invariants are studied the cauchy problem for these equations is solved by an analog of the inverse scattering problem method

dressing operator approach to the toroidal model of higher dimensional dispersionless kp hierarchy based upon the dressing operator approach we investigate the twistor theoretical construction additional symmetries hodograph solutions and miura transformation for the toroidal model of higher dimensional dispersionless kp hierarchy introduced by takasaki

the brezis nirenberg problem for the curl curl operator abstract we look for solutions e r 3 of the problem e e e p 2 e in e 0 on on a bounded lipschitz domain r 3 where denotes the curl operator in r 3 the equation describes the propagation of the time harmonic electric field e x e i t in a nonlinear isotropic material with e 2 0 where and e stand for the permeability and the linear part of the permittivity of the material the nonlinear term e p 2 e with p 2 is responsible for the nonlinear polarisation of and the boundary conditions are those for surrounded by a perfect conductor the problem has a variational structure and we deal with the critical value p for instance in convex domains or in domains with c 1 1 boundary p 6 2 is the sobolev critical exponent and we get the quintic nonlinearity in the equation we show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on 0 we develop a new critical point theory which allows to solve the problem and which enables us to treat more general anisotropic media as well as other variational problems

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

additional symmetries and solutions of the dispersionless kp hierarchy the dispersionless kp hierarchy is considered from the point of view of the twistor formalism a set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied a method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed this method is applied for determining the orbits of solutions satisfying reduction constraints of gelfand dikii type under the action of additional symmetries

critical growth problems for polyharmonic operators we prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower order perturbations of the critical term the results highlight the phenomenon of bifurcation of the critical dimensions discovered by pucci and serrin moreover we show that another bifurcation seems to appear for nonresonant dimensions

the nehari manifold and the existence of multiple solutions for a singular quasilinear elliptic equation abstract in this paper we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem div x a p u p 2 u h x u m 2 u h x u n 2 u x u x 0 x where r n n 3 is a bounded domain with smooth boundary 0 1 p n 0 a n p p 1 m p n p n n 1 a p 0 h x h x are lebesgue measurable functions which may change sign on we prove that there exist at least two positive solutions by using the nehari manifold and the fibrering maps associated with the energy functional for this problem

subcritical perturbations of a singular quasilinear elliptic equation involving the critical hardy sobolev exponent abstract in this work we improve some known results for a singular operator and also for a wide class of lower order terms by proving a multiplicity result the proof is made by applying the generalized mountain pass theorem due to ambrosetti and rabinowitz to do this we show that the minimax levels are in a convenient range by combining a special class of approximating functions due to gazzola and ruf with the concentrating functions of the best sobolev constant

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

functional symmetries and solutions of the dkp hierarchy it is shown that the dispersionless kp hierarchy possesses an infinite collection of symmetries depending on arbitrary functions of the time variables the twistor formulation of a factorization problem determines the family of symmetries and serves for the characterization of solutions of the hierarchy

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

multiple constant sign and nodal solutions for nonlinear elliptic equations with the p laplacian we consider a nonlinear elliptic equation driven by the p laplacian with dirichlet boundary conditions using variational techniques combined with the method of upper lower solutions and suitable truncation arguments we establish the existence of at least five nontrivial solutions two positive two negative and a nodal sign changing solution our framework of analysis incorporates both coercive and p superlinear problems also the result on multiple constant sign solutions incorporates the case of concave convex nonlinearities

p adic l functions of bianchi modular forms the theory of overconvergent modular symbols developed by rob pollack and glenn stevens gives a beautiful and effective construction of the p adic l function of a modular form in this paper we give an analogue of their results for bianchi modular forms that is modular forms over imaginary quadratic fields in particular we prove control theorems that say that the canonical specialisation map from overconvergent to classical bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable hecke operators we also give an explicit link between the classical modular symbol attached to a bianchi modular form and critical values of its l function which then allows us to construct p adic l functions of bianchi modular forms

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

computing a lower bound for the canonical height on elliptic curves over number fields computing a lower bound for the canonical height is a crucial step in determining a mordell weil basis for elliptic curves this paper presents an algorithm for computing such a lower bound for elliptic curves over number fields without searching for points the algorithm is illustrated by some examples

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

semilinear elliptic equations and supercritical growth where p p the aim of the present paper is to understand for the case p pc the behaviour of solutions to 1 i with large supremum norm to this end 0 is restricted to be the unit ball f3c r3 as in this case the behaviour of solutions to 1 1 can be described in some detail according to the results proved in gidas ni and nirenberg s all positive solutions of 1 1 must be radial when q b with the scaling iid ilip in the case n 3 problem 1 1 reduces to the following ordinary differential equation problem

on the quasilinear elliptic problem with a critical hardy sobolev exponent and a hardy term abstract in the present paper a quasilinear elliptic problem with a critical sobolev exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions for the problem is obtained the result depends crucially on the parameters p t s and

periods of cusp forms and elliptic curves over imaginary quadratic fields in this paper we explore the arithmetic correspondence between on the one hand isogeny classes of elliptic curves e defined over an imaginary quadratic field k of class number one and on the other hand rational newforms f of weight two for the congruence subgroups where n is an ideal in the ring of integers r of k this continues work of the first author and forms part of the ph d thesis of the second author in each case we compute numerically the value of the l series at and compare with the value of which is predicted by the birch swinnerton dyer conjecture finding agreement to several decimal places in particular we find that whenever has a point of infinite order several examples are given in detail from the extensive tables computed by the authors

the cauchy problem for the pavlov equation with large data abstract we prove a local solvability of the cauchy problem for the pavlov equation with large initial data by the inverse scattering method the pavlov equation arises in studies einstein weyl geometries and dispersionless integrable models our theory yields a local solvability of cauchy problems for a quasi linear wave equation with a characteristic initial hypersurface

on the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields it was shown recently that frobenius reduction of the matrix fields reveals interesting relations among the nonlinear partial differential equations pdes integrable by the inverse spectral transform method s integrable pdes linearizable by the hopf cole substitution c integrable pdes and integrable by the method of characteristics ch integrable pdes however only two classes of s integrable pdes have been involved soliton equations such as korteweg de vries nonlinear schrodinger kadomtsev petviashvili and davey stewartson equations and gl n c self dual type pdes such as yang mills equation in this paper we consider the simple five dimensional nonlinear pde from another class of s integrable pdes namely scalar nonlinear pde which is the commutativity condition of the pair of vector fields we show its origin from the 1 1 dimensional hierarchy of ch integrable pdes after certain composition of frobenius type and differential reductions imposed on the matrix fields matrix generaliza

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

ground state solutions for singular p laplacian equation in rn abstract in this work we study the ground state solution for the class of singular quasilinear elliptic problem of the form div x a p u p 2 u h x u m 2 u h x u q 2 u x r n where 1 p n a n p p and h r h r 0 we prove the compactly embedding results of the weighted z d a r 1 p r n l h r m r n space of radially symmetric functions and then obtain the existence of nontrivial ground state solution

on the whitham hierarchy dressing scheme string equations and additional symmetries a new description of the universal whitham hierarchy in terms of a factorization problem in the lie group of canonical transformations is provided this scheme allows us to give a natural description of dressing transformations string equations and additional symmetries for the whitham hierarchy we show how to dress any given solution and prove that any solution of the hierarchy may be undressed and therefore comes from a factorization of a canonical transformation a particularly important function related to the tau function appears as a potential of the hierarchy we introduce a class of string equations which extends and contains previous classes of string equations considered by krichever and by takasaki and takebe the scheme is also applied for a convenient derivation of additional symmetries moreover new functional symmetries of the zakharov extension of the bentley gas equations are given and the action of additional symmetries over the potential in terms of linear pdes is characterized

darmon points on elliptic curves over number fields of arbitrary signature we present new constructions of complex and pp adic darmon points on elliptic curves over base fields of arbitrary signature we conjecture that these points are global and present numerical evidence to support our conjecture

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

existence of nontrivial solutions for singular quasilinear elliptic equations on r n in this paper we study the existence of weak solutions for the singular quasilinear elliptic problem 0 1 div x a p u p 2 u v x u p 2u h x u s 2u h x u r 2u x r n u x 0 as x where 1

existence of multiple solutions for an elliptic system with sign changing weight functions abstract in this paper we will consider a class of quasilinear elliptic problem of the form d i v x a p u p 2 u g 1 x u p 2 u h x u 2 u v h 1 x u n 2 u d i v x a p v p 2 v g 2 x v p 2 v h x v 2 v u h 2 x v n 2 v u x 0 v x 0 x r n where 0 1 p n 1 n p p n p n p d 0 a n p p a b a 1 d a 1 b 0 the weight g 1 x g 2 x are bounded and nonnegative functions and h x h 1 x h 2 x are continuous functions which change sign in r n we will prove that the problem has at least two positive solutions by using the nehari manifold and the fibering maps associated with the euler function for this problem

positive solutions to the weighted critical quasilinear problems in a bounded domain with smooth boundary we consider a kind of weighted quasilinear elliptic problem which satisfies dirichlet boundary condition and involves the hardy sobolev inequality by the analytic techniques we first get the properties of the extremal functions by which the best hardy sobolev constant is achieved then by the variational methods the existence of positive solutions to the problem is verified by careful estimates and computations

doubly cuspidal cohomology for principal congruence subgroups of gl 3 z the cohomology of arithmetic groups is made up of two pieces the cuspidal and noncuspidal parts within the cuspidal cohomology is a subspacethe f cuspidal cohomology spanned by the classes that generate representations of the associated finite lie group which are cuspidal in the sense of finite lie group theory few concrete examples of f cuspidal cohomology have been computed geometrically outside the cases of rational rank 1 or where the symmetric space has a hermitian structure this paper presents new computations of the f cuspidal cohomology of principal congruence subgroups f p of gl 3 2 of prime level p we show that the f cuspidal cohomology of f p vanishes for all p 19 with p 11 but that it is nonzero for p 11 we give a precise description of the f cuspidal cohomology for jt 11 in terms of the f cuspidal representations of the finite lie group gl 3 e 11 we obtained the result ultimately by proving that a certain large complex matrix m is rank deficient computation with the svd algorithm gave strong evidence that m was rank deficient but to prove it we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields

on the rth dispersionless toda hierarchy factorization problem additional symmetries and some solutions for a family of poisson algebras parametrized by and an associated lie algebraic splitting we consider the factorization of given canonical transformations in this context we rederive the recently found rth dispersionless modified kp hierarchies and rth dispersionless dym hierarchies giving a new miura map among them we also found a new integrable hierarchy which we call the rth dispersionless toda hierarchy moreover additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless kp rth dispersionless dym and rth dispersionless toda equations some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless dym equation for which a complete integral is presented and therefore an appropriate envelope leads to a general solution symmetries and miura maps are applied to get new solutions and solutions of the rth dispersionless modified kp equation