on meromorphic solutions of nonlinear partial differential equations of first order abstract we prove a uniqueness theorem in terms of value distribution for meromorphic solutions of a class of nonlinear partial differential equations of first order which shows that such solutions f are uniquely determined by the zeros and poles of f c j counting multiplicities for two distinct complex numbers c 1 and c 2

on the question of whether f e zf b z f 0 can admit a solution f 0 of finite order on montre que si b z est soit i une fonction entiere transcendante d ordre 1 soit ii un polynome de degre impair alors toute solution f 0 de l equation f e z f b z f 0 est d ordre infini on obtient un resultat partiel quand b z est un polynome de degre pair

critical phenomena in linear elliptic problems abstract in this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation u v x u in r n and u d 1 2 r n where n 3 and v l 1 loc r n with v 0 in r n we establish that when potential v has strong singularities phenomena which appear under critical exponent nonlinearities brezis and nirenberg are present in linear problems as well

a global branch of positive solutions above the continuous spectrum for problems with indefinite nonlinearities we prove the existence and bifurcation of a global branch of positive solutions for a nonlinear neumann eigenvalue problem on the half axis 0 the nonlinearity is assumed to have a superlinear growth multiplied by a weight function changing sign this leads to the existence of nontrivial solutions above the continuous spectrum of the linearised problem

the chiral potts model and its associated link invariant a new link invariant is derived using the exactly solvable chiral potts model and a generalized gaussian summation identity starting from a general formulation of link invariants using edge interaction spin models we establish the uniqueness of the invariant for self dual models we next apply the formulation to the self dual chiral potts model and obtain a link invariant in the form of a lattice sum defined by a matrix associated with the link diagram a generalized gaussian summation identity is then used to carry out this lattice sum enabling us to cast the invariant into a tractable form the resulting expression for the link invariant is characterized by roots of unity and does not appear to belong to the usual quantum group family of invariants a table of invariants for links with up to eight crossings is given

the possible orders of growth of solutions to certain linear differential equations near a singular point abstract in this paper we give the possible orders of growth of solutions to certain linear differential equations near a finite singular point for that an asymptotic equality of wiman valiron type is proved for the derivatives of functions which are analytic in the closed complex plane except at a finite singular point

on a class of degenerate and singular elliptic systems in bounded domains abstract this paper deals with the nonexistence and multiplicity of nonnegative nontrivial solutions to a class of degenerate and singular elliptic systems of the form div h 1 x u f u x u v in div h 2 x v f v x u v in where is a bounded domain with smooth boundary in r n n 2 and h i 0 h i l loc 1 h i i 1 2 are allowed to have essential zeroes at some points in f u f v f and is a positive parameter our proofs rely essentially on the critical point theory tools combined with a variant of the caffarelli kohn nirenberg inequality in p caldiroli r musina on a variational degenerate elliptic problem nodea nonlinear differential equations appl 7 2000 189 199

on solutions of linear differential equations with entire coefficients abstract the first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions subsequent results involve the separation of the zeros of a bank laine function

the abc theorem for higher dimensional function fields we generalize the abc theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non singular in codimension one we also obtain an upper bound for the minimal order sequence of wronskians over such function fields of positive characteristic

multiple positive solutions for semilinear dirichlet problems with sign changing weight function in infinite strip domains abstract in this paper existence and multiplicity results to the following dirichlet problem u u f x u q 1 h x u p 1 in u 0 in u 0 on are established where r r n 1 is bounded smooth domain and n 2 here 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 is a positive real parameter the function f among other conditions can possibly change sign in and the function h satisfies suitable conditions the study is based on the comparison of energy levels on nehari manifold

perturbed kirchhoff type neumann problems in orlicz sobolev spaces the aim of this paper is to establish the existence of infinitely many solutions for perturbed kirchhoff type non homogeneous neumann problems involving two parameters to be precise we prove that an appropriate oscillating behaviour of the nonlinear term even under small perturbations ensures the existence of infinitely many solutions our approach is based on recent variational methods for smooth functionals defined on orlicz sobolev spaces

on a p biharmonic problem with no flux boundary condition the study of fourth order partial differential equations has flourished in the last years however a p biharmonic problem with no flux boundary condition has never been considered before not even for constant p this is an important step further since surfaces that are impermeable to some contaminants are appearing quite often in nature hence the significance of such boundary condition by relying on several variational arguments we obtain the existence and the multiplicity of weak solutions to our problem we point out that although we use a mountain pass type theorem in order to establish the multiplicity result we do not impose an ambrosettirabinowitz type condition nor a symmetry condition on our nonlinearity f

finite order solutions of second order linear differential equations on considere l equation differentielle f a z f b z f 0 ou a z et b z sont des fonctions entieres on trouve des conditions sur a z et b z qui garantissent que chaque solution f 0 de l equation a un ordre fini

non archimedean nevanlinna theory in several variables and the non archimedean nevanlinna inverse problem cartan s method is used to prove a several variable non archimedean nevanlinna second main theorem for hyperplanes in projective space the corresponding defect relation is derived but unlike in the complex case we show that there can only be finitely many non zero non archimedean defects we then address the non archimedean nevanlinna inverse problem by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space there exists a non archimedean analytic function with the given defects at the specified hyperplanes and with no other defects 1 history and introduction nevanlinna theory broadly speaking studies to what extent something like the fundamental theorem of algebra holds for meromorphic functions unlike polynomials transcendental meromorphic functions in general have infinitely many zeros however they have only finitely many zeros inside a disc of radius r therefore in order to study the values of a meromorphic function nevanlinna theory associates to each meromorphic function f three functions of r the distance from the origin for their precise definitions see nevanlinna ne 2 the characteristic or height function tf r measures the growth of f and should be thought of as the analogue of the degree of a polynomial the counting function nf a r counts the number of times as a logarithmic average f takes on the value a in the disc of radius r finally the mean proximity function mf a r measures how often on average f stays close to the value a on the circle of radius r nevanlinna proved two main theorems about these functions the so called first main theorem states that tf r mf a r nf a r o 1 where the bounded term o 1 depends on f and a but not on r this should be thought of as a substitute for the fundamental theorem of algebra in the following sense the first main theorem says that mf a r nf a r is essentially independent of the value a and this is analogous to the fact that a polynomial takes on every finite value the same number of times counting multiplicity the first received by the editors october 14 1995 and in revised form june 17 1996 1991 mathematics subject classification primary 11j99 11s80 30d35 32h30 32p05

oscillation results of certain higher order linear differential equations with periodic coefficients in the complex plane abstract 1 has given a characterization of homogeneous linear differential equations with certain analytic periodic coefficients which admits a solution with finite exponent of convergence however her method seems too general and in most cases too complicated for applications we give in this paper a direct approach to the problem and obtain several such characterizations which do not seem to follow from those of baesch in particular an explicit necessary and sufficient condition to the problem is given for certain third order equations the results again do not seem to follow from those of baesch our method is based on that of 8 which in turn depends on basic representations of solutions given by bank laine and langley

on the abc conjecture and diophantine approximation by rational points we show that an earlier conjecture of the author on diophantine approximation of rational points on varieties implies the abc conjecture of masser and oesterle in fact a weak form of the former conjecture is sufficient involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type analogues of this weaker conjecture are proved in the split function field case of characteristic zero and in the case of holomorphic curves nevanlinna theory the proof of the latter involves a geometric generalization of the classical lemma on the logarithmic derivative due to mcquillan this lemma may be of independent interest

the possible orders of solutions of linear differential equations with polynomial coefficients we find specific inforrnation about the possible orders of transeendental solutions of equations of the form f n pn l z f n l po z f0 where po z pl z apnl z are polynomials with po z g 0 several examples are given

logarithmic derivatives in annuli in this paper we define nevanlinna functions in annuli with two independent variables and use them to prove a version of the generalized logarithmic derivative lemma in annuli as a corollary we can conclude the logarithmic derivative lemma in a disk and the complex plane

zeros of analytic functions with or without multiplicities the classical mason stothers theorem deals with nontrivial polynomial solutions to the equation a b c it provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a deg b and deg c we extend this to general analytic functions living on a reasonable bounded domain omega subset mathbb c rather than on the whole of mathbb c the estimates obtained are sharp for any and a generalization of the original result on polynomials can be recovered from them by a limiting argument

a generalization of torres second relation let l kt u u aj be a tame link in s3 of ii 2 components and let l be its sublink ls l k let h and hh be the abelianizations of w s3 l and w s3 z m respectively and let r m resp z zi j is the unique ring homomorphism with f for 1 2 j f l ek l q ek l q et lj and examples are given to indicate that either of these inclusions may be an equality this theorem is used to generalize certain known properties of ex to the higher ideals

an analogue of continued fractions in number theory for nevanlinna theory we show an analogue of continued fractions in approximation to irrational numbers by rationals for nevanlinna theory the analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of nevanlinna theory

multiplicity of positive solution for an indefinite elliptic equation on r2 with exponential nonlinearities abstract in this paper we consider the multiplicity of positive solution to the equation u u h x u p e u x r 2 with h x a sign changing function p 1 a constant and a parameter we first use a moving plane argument to get a priori bounds for the positive solutions of this equation then we obtain multiple positive solutions through a squeezing method which overcomes the lack of compactness of the problem

a concentration compactness lemma with applications to singular eigenvalue problems let omega subset of or equal to r n be any open set we study the nonlinear eigenvalue problem delta p u lambda v x u p 2 u u epsilon d 0 1 p omega where 1 p n and v is an element of l loc 1 omega may have strong singularities and an indefinite sign the key ingredient is a precised concentration compactness lemma related to v dependent limiting problems this work follows extends and simplifies that of a tertikas 1998 j funct anal 154 42 66 dealing with the positive linear case for omega r n c 1999 academic press

a note on the abc conjecture in this note we have generalized the abc conjecture for integers and extended mason s theorem to polynomials in m by dint of nevanlinna s value distribution theory 2002 wiley periodicals inc

on the dependent property of solutions for general higher order periodic differential equation abstract in this paper the property of linearly dependence of solutions f z and f z 2 i for higher order linear differential equation f k z a k 2 z f k 2 z a 0 z f z 0 where a j z j 0 k 2 are entire periodic coefficients and a s z s 2 k 2 is the dominant coefficient is investigated

linking numbers and the elementary ideals of links let l kx u u k c s3 be a tame link of ii 2 components and h the abelianization of g w s3 l let be the ii x ii matrix with entries in zh given by h lk il k kk tk and for i j t

change of stability for schrodinger semigroups when the parameter x varies in the positive real axis here m is some bounded and continuous weight function being positive somewhere and u0 an initial condition the stability is understood as stability in the norm we point out that by duality and interpolation we also get the stability in lp r 1 g p oo the question of stability of the zero solution is closely related to the existence of a principal eigenvalue for the elliptic eigenvalue problem

on nevanlinna s second main theorem in projective space we first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables then we show the best nature of the second main theorem in nevanlinna theory under two different assumptions of non degeneracy of meromorphic mappingsf n n for arbitrary positive integersn andm moreover we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order finally we demonstrate the sharpness of all upper bounds in our main theorems

remarks on a brezis nirenberg s result in this note we revisit the equation with sobolev critical exponent which is firstly studied by brezis nirenberg in 4 p u u u p 2u u 2 2u in u h01 where rn is a bounded smooth domain n 3 2 0 r we show that for any m n there exists m 0 such that p has m solutions for any r 0 and every 0 m when n 4 0 we show that there is 0 0 such that problem p has a nontrivial solution for every 0 moreover we get multiple sign changing solutions for problem p

global bifurcation results for a semilinear elliptic equation on all of mathbb r n we prove the existence of positive solutions for the semilinear elliptic equation u x g x f u x 0 u 1 for x r lim x u x 0 which arises in population genetics under the hypotheses that n 3 4 5 and g lies in l n 2 ir n we establish the existence of a principal eigenvalue 1 for the corresponding linearized problem and making use of the asymptotic properties of solutions and local and global bifurcation theory prove the existence of a continuum of solutions lying in the space d 1 2 extending from 1 to

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

the fibering map approach to a nonlocal problem involving p x laplacian abstract we study the existence of positive solutions for a p x kirchhoff problem the main tool used is the fibering map approach for the corresponding nehari manifold

milnor s invariants and the completions of link modules let l be a tame link of k 2 components in s3 h the abelianization of its group 7tj s3 l and ih the augmentation ideal of the integral group ring zh the ih adic completions of the alexander module and alexander invariant of l are shown to possess presentation matrices whose entries are given in terms of certain integers t4 i1 iq introduced by j milnor various applications to the theory of the elementary ideals of these modules are given including a condition on the alexander polynomial necessary for the linking numbers of the components of l with each other to all be zero in the special case 2 it is shown that the various milnor invariants f r 1 s 1 are determined up to sign by the alexander polynomial of l and that this alexander polynomial is 0 iff i r 1 s 1 0 for all r s 0 with r s even also the chen groups of l are determined up to isomorphism by those nonzero i r 1 s 1 with r s minimal in contrast it is shown by example that for k 3 the alexander polynomials of a link and its sublinks do not determine its chen groups

representations of solutions of periodic second order linear differential equations on considere l equation y g z y h z y 0 a coefficients entiers periodiques arbitraires g z et h z de periode 0 et des solutions y z ayant la propriete que l exposant de convergence de la suite nulle est finie

composite bank laine functions and a question of rubel a bank laine function is an entire function e satisfying e z 1 at every zero of e we determine all bank laine functions of form e fog with f g entire further we prove that if h is a transcendental entire function of finite order then there exists a path tending to infinity on which h and all its derivatives tend to infinity thus establishing for finite order a conjecture of rubel

multiple perturbations of a singular eigenvalue problem abstract we study the perturbation by a critical term and a p 1 superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential by means of variational arguments and a version of the concentration compactness principle in the singular case we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem

the application of value distribution theory to a doubly anharmonic oscillator the model of doubly anharmonic oscillators is first transformed into certain periodic second ordinary differential equations a class of exact solutions for eigenfunctions and eigenvalues is obtained from bank and laine s theory on periodic ordinary differential equations which is expressed in the form of the products of the polynomial and exponential functions when parameters satisfy some special relations

liouville type result and long time behavior for fisher kpp equation with sign changing and decaying potentials abstract this paper concerns the liouville type result for the general semilinear elliptic equation s a i j x i j u x k q i x i u x f x u x 0 a e in r n where f is of the kpp monostable nonlinearity as a continuation of the previous works of the second author 31 32 the novelty of this work is that we allow f s x 0 to be sign changing and to decay fast up to a hardy potential near infinity first we introduce a weighted generalized principal eigenvalue and use it to characterize the liouville type result for eq s that was proposed by h berestycki secondly if a i j is the identity matrix and q is a compactly supported divergence free vector field we find a condition that eq s admits no positive solution for k k where k is a certain positive threshold to achieve this we derive some new techniques thanks to the recent results on the principal spectral theory for elliptic operators 14 to overcome some fundamental difficulties arising from the lack of compactness in the domain this extends a nice result of berestycki hamel nadirashvili 5 on the limit of eigenvalues with large drift to the case without periodic condition lastly the well posedness and long time behavior of the evolution equation corresponding to s are further investigated the main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for s holds or not

multiple positive solutions for a semilinear equation with prescribed singularity variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points some exact local behavior for positive solutions are also given

critical singular problems on infinite cones abstract we prove existence results for non autonomous perturbations of critical singular elliptic boundary value problems the non singular case was treated by tarantello ann inst h poincare analyse non lineaire 9 1992 281 for bounded domains here the singular weight allows for unbounded domains as cones and give rise to a different non compactness picture as was first remarked by caldiroli and musina calc variations pde 8 1999 365

principal eigenvalues with indefinite weight functions both existence and non existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved the existence of a continuous family of principal eigenvalues is demonstrated

a sharp form of nevanlinna s second fundamental theorem letf be meromorphic in the plane we find a sharp upper bound for the error term s r f m r f sum limits i 1 q m r a i f n 1 r f 2t r f in nevanlinna s second fundamental theorem for any positive increasing functions t t andp t with int limits 1 infty dt varphi t infty and int limits 1 infty dt p t infty we have s left r f right leqq log left frac varphi left t left r f right right p left r right right o left 1 right asr outside a sete with int limits e dr p r infty further if t t is positive and increasing and int limits 1 infty dt psi t infty then there is an entiref such thats r f log t r f outside a set of finite linear measure we also prove analogous results for functions meromorphic in a disk

existence and asymptotic behavior of ground states for quasilinear singular equations involving hardy sobolev exponents abstract we study the existence and decaying rate of solutions for the quasilinear problem p u x f u x g u in r n u 0 in r n u x 0 where p stands for the p laplacian operator 1 p n r n 0 is continuous and not identically zero 0 is a parameter x is the euclidean norm of x 0 p f g 0 0 are continuous and nondecreasing f has sublinear growth and the hardy sobolev exponent p p n n p bounds the growth of g we deal with variational methods and the lower and upper solutions technique

on cartan s theorem and cartan s conjecture we give a mild generalization of cartan s theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes this generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author given hyperplanes in projective space in general position there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union even linearly degenerate curves satisfy the inequality of cartan s theorem except for the ramification term in addition it is shown how these methods can lead to a shorter proof of nochka s theorem on cartan s conjecture and in the number field case how nochka s theorem gives a short proof of wirsing s theorem on approximation of algebraic numbers by algebraic numbers of bounded degree