existence of infinitely many solutions to a class of kirchhoff schrodinger poisson system in this paper we consider the existence of infinitely many solutions to following nonlinear kirchhoff schrodinger poisson system a b r 3 u 2 v x u 2 u v x u l x u f x u x r 3 l x u 2 x r 3 where constants a 0 b 0 and 0 when f has sublinear growth in u we obtain infinitely many solutions under certain assumption that v do not have a positive lower bound the technique we use in this paper is the symmetric mountain pass theorem established by kajikiya 2005

solitons and the electromagnetic field in a recent paper 4 it has been introduced a lorentz invariant equation in three space dimensions having soliton like solutions we recall that roughly speaking a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations see 6 15 13 10 the equation introduced in 4 is the euler lagrange equation of an action functional

single and multi peak solutions for a nonlinear maxwell schrodinger system with a general nonlinearity abstract for the following elliptic system in r 3 e 2 v v x v x v f v v 2 lim x x 0 we construct positive solutions u and which concentrate at several given isolated local minimum components of v as e 0 the potential v is a strictly positive continuous function and the nonlinearity f is subcritical near infinity and superlinear near zero and satisfies only the berestycki lions condition

generalizations of laguerre polynomials it is shown that the polynomials ln m0 m1 mn x n 0 defined by ln m0m1 mn x k 0n 1ak dkln x for certain real coefficients ak k 0n 1 are orthogonal with respect to the inner product f g 1 1 0 x e x f x g x dx v 0nmv f v 0 g v 0 where 1 n n and mv 0 for all v 0 1 2 n for these new polynomials n m0 m1 mn x n 0 an orthogonality relation and a second order differential equation are derived further we obtain a representation as a n 2fn 2 hypergeometric series and a 2n 3 terms recurrence relation which gives rise to a christoffel darboux type formula

linear perturbations of differential or difference operators with polynomials as eigenfunctions abstract this paper deals with one parameter linear perturbations of a family of polynomials pn x n 0 with deg pn x n of the form p n x p n x q n x where is a real parameter and qn x n 0 are polynomials with deg qn x n let the polynomials pn x n 0 be eigenfunctions of a linear differential or difference operator l a with eigenvalues n n 0 the purpose of this p to derive necessary and sufficient conditions for the polynomials qn x n 0 such that the polynomials pn x n 0 are eigenfunctions of a linear difference or differential operator possibly of infinite order of the form l a with eigenvalues n n n 0

differential and difference operators having orthogonal polynomials with two linear perturbations as eigenfunctions abstract in this paper we consider the polynomials pn x n 0 orthogonal with respect to a certain symmetric bilinear form of sobolev type these polynomials are the result of two linear perturbations to the orthogonal polynomials pn x n 0 eigenfunctions of a linear differential or difference operator l we show that the polynomials pn x n 0 are eigenfunctions of one or more linear differential or difference operators possibly of infinite order of the form l a b c

a note on the koekoek s differential equation for generalized jacobi polynomials abstract in a recent paper differential equations for generalized jacobi polynomials submitted for publication koekoek and koekoek discovered a linear differential equation for the polynomials pn m n x n 0 which are orthogonal on 1 1 with respect to 0 1 2 2 1 1 1 1 x 1 x m x 1 n x 1 1 m n 0 this differential equation is of infinite order except in a number of cases it is the purpose of this note to reprove and interpret the results of the koekoeks in the finite order cases in a short and easy way

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

self adjoint differential equations for classical orthogonal polynomials this paper deals with spectral type differential equations of the self adjoint differential operator 2r order l 2r y x 1 x dr dxr x r x dry x dxr rny x if x is the weight function and x is a second degree polynomial function then the corresponding classical orthogonal polynomials qn x n 0 are shown to satisfy this differential equation when rn is given by rn r 1k 0 n k 1 n k 1 2 where 1 and 2 are the leading coefficients of the two polynomial functions associated with the classical orthogonal polynomials moreover the singular eigenvalue problem associated with this differential equation is shown to have qn x and rn as eigenfunctions and eigenvalues respectively any linear combination of such self adjoint operators has qn x as eigenfunctions and the corresponding linear combination of rn as eigenvalues

differential equations of infinite order for sobolev type orthogonal polynomials abstract assume that pn x n 0 are orthogonal polynomials relative to a quasi definite moment functional which satisfy a differential equation of spectral type of order d 2 d where l i x are polynomials of degree i let be the symmetric bilinear form of discrete sobolev type defined by p q pq np k c q k c where n 0 and c are real constants k is a non negative integer and p and q are polynomials we first give a necessary and sufficient condition for to be quasi definite and then show if is quasi definite then the corresponding sobolev type orthogonal polynomials rnn k c x n 0 satisfy a differential equation of infinite order of the form p q np k c q k c where ai x i 0 are polynomials of degree i independent of n except a0 x a0 x n we also discuss conditions 3nder which such a differential equation is of finite order when is positive definite d

the search for differential equations for certain sets of orthogonal polynomials abstract we look for differential equations of the form i 0ci x y i x 0 where the coefficients ci x i 0 are continuous functions on the real line and where ci x i 1 are independent of n for the generalized jacobi polynomials p m nn x n 0 and for generalized laguerre polynomials l m nn x n 0 which are orthogonal with respect to an inner product of sobolev type we use a method involving computer algebra packages like maple and mathematica and we will give some preliminary results

using d operators to construct orthogonal polynomials satisfying higher order q difference equations abstract let p n n be either the q meixner or the q laguerre polynomials we form a new sequence of polynomials q n n by considering a linear combination of two consecutive p n q n p n n p n 1 n r using the concept of d operator we generate sequences n n for which the polynomials q n n are orthogonal with respect to a measure and common eigenfunctions of a higher order q difference operator

partial differential equations having orthogonal polynomial solutions abstract we show that if a second order partial differential equation l u auxx 2buxy cuyy dux euy nu has orthogonal polynomial solutions then the differential operator l must be symmetrizable and can not be parabolic in any nonempty open subset of the plane we also find rodrigues type formula for orthogonal polynomial solutions of such differential equations

differential operators having sobolev type jacobi polynomials as eigenfunctions in a recent paper koekoek and koekoek j comput appl math 126 2000 1 31 discovered a linear differential equation for the jacobi type polynomials pn m n x n 0 which are orthogonal on 1 1 with respect to 2 2 1 1 1 1 x 1 x m x 1 n x 1 1 m n 0 0 1 if m2 n2 0 this differential equation is of finite order in the following cases 1 m 0 n 0 and 0 1 2 2 m 0 n 0 and 0 1 2 3 m 0 n 0 and 0 1 2 in this paper the result will be generalized to sobolev type jacobi polynomials

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

differential equations for generalized jacobi polynomials abstract we look for differential equations of the form m i 0 a i x y i x n i 0 b i x y i x mn i 0 c i x y i x 1 x 2 y x 2 x y x n n 1 y x 0 satisfied by the generalized jacobi polynomials pn m n x n 0 which are orthogonal on the interval 1 1 with respect to the weight function 2 2 1 1 1 1 x 1 x m x 1 n x 1 where 1 1 m 0 and n 0 we give explicit representations for the coefficients ai x i 0 bi x i 0 and ci x i 0 and we show that this differential equation is uniquely determined for m2 n2 0 the order of this differential equation is infinite except for 0 1 2 or 0 1 2 moreover the order equals 2 4 if m 0 n 0 and 0 1 2 2 4 if m 0 n 0 and 0 1 2 2 2 6 if m 0 n 0 and 0 1 2

positive solutions for schrodinger poisson equations with a critical exponent abstract in this paper we establish the existence of a positive solution of the schrodinger poisson equations with a critical sobolev exponent the methods used here are based on the concentration compactness principle of p l lions and methods of brezis and nirenberg

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

distributional solutions of the hypergeometric differential equation abstract we present the distributional solutions to the hypergeometric differential equation these solutions are obtained in the form of infinite series of the dirac delta functions and its derivatives we employ these solutions to observe their interesting features furthermore the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials

existence of infinitely many large solutions for the nonlinear schrodinger maxwell equations abstract in this paper we study the nonlinear stationary schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 using the variant fountain theorem introduced by zou w zou variant fountain theorems and their applications manuscripta math 104 2001 343 358 under certain assumptions on v and f we get infinitely many large solutions for

singular second order operators the maximal and minimal operators and selfadjoint operators in between differential operators arising from the differential equation py qy lambda wy are put in the hilbert space setting l 2 a b w a new and simpler characterization of the minimal operator is exhibited selfadjoint operators which lie in between the minimal and maximal operator are easily described in terms of conditions on boundary coefficients which look like and indeed are the same as those imposed on regular problems examples drawn from mathematical physics include the legendre laguerre hermite and bessel problems

some functions that generalize the krall laguerre polynomials let l alpha be the semi infinite tridiagonal matrix associated with the three term recursion relation satisfied by the laguerre polynomials with weight function 1 gamma alpha 1 z alpha e z alpha 1 on the interval 0 infinity we show that when alpha is a positive integer by performing at most alpha successive darboux transformations from l alpha we obtain orthogonal polynomials on 0 infinity with weight distribution 1 gamma alpha k 1 z alpha k e z sigma j 1 k s j delta k j z with 1 less than or equal to k less than or equal to alpha we prove that as a consequence of the rational character of the darboux factorization these polynomials are eigenfunctions of a finite order differential operator our construction calls for a natural bi infinite extension of these results with polynomials replaced by functions of which the semi infinite case is a limiting situation c 1999 elsevier science b v all rights reserved

solutions of a schrodinger poisson system with combined nonlinearities abstract in this paper we are concerned with the following schrodinger poisson system u v x u u k x u q 2 u f x u x r 3 u 2 lim x x 0 x r 3 where 0 is a parameter 1 q 2 and f x u is linearly bounded in u at infinity we establish the existence and multiplicity of solutions under suitable assumptions on v k and f

differential operators having symmetric orthogonal polynomials as eigenfunctions abstract let the polynomials pn x n 0 orthogonal with respect to a symmetric positive definite moment functional be eigenfunctions of a linear differential operator l we consider the orthogonal polynomials pn x n 0 and pn v x n 0 which are obtained by adding one resp two symmetric sobolev type terms to in all the cases we derive a representation for the polynomials and show that they are eigenfunctions of one or more linear differential operators mostly of infinite order of the form l a resp l a v b v c further it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues

existence and multiplicity of non trivial solutions for a class of modified schrodinger poisson systems abstract in the present paper we study the following modified schrodinger poisson system p u v x u u 1 2 u u 2 u p 1 u h x in r 3 u 2 in r 3 where 4 p 1 12 h l 2 r 3 and h 0 this model describes the interaction of a charge particle with an electro magnetic field under some hypotheses on v x we prove the existence of a non trivial ground state solution and two non trivial ground state solutions for the system p

infinitely many solutions for a class of sublinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many solutions for a class of sublinear schrodinger maxwell equations the proof is based on the variant fountain theorem established by zou recent results from the literature are extended

multiple solutions for a nonlinear schrodinger poisson system with sign changing potential this paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear schrodinger poisson system under some suitable conditions some criteria on the existence of nontrivial solutions including a ground state solution two solutions and a sequence of nontrivial solutions u k with u k 0 as k is established by applying the morse theory and variational methods some known results in the literatures are extended and improved

high energy solutions for the superlinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many large energy solutions for the superlinear schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 via the fountain theorem in critical point theory

on ground state solutions for the schrodinger poisson equations with critical growth abstract in this paper we study the existence of ground state solutions for the following schrodinger poisson equation u v x u u u q 1 u u 4 u in r 3 u 2 in r 3 where is a positive parameter under some certain assumptions on v we prove that for every 0 and q 2 5 such a class of schrodinger poisson equation with critical growth has at least a positive ground state solution via variational methods some recent results from the literature are extended

from krall discrete orthogonal polynomials to krall polynomials abstract we show how to get krall polynomials from krall discrete polynomials using a procedure of passing to the limit in some of the parameters of the family we also show that this procedure has to be different to the standard one used in the askey scheme to go from the classical discrete polynomials to the classical polynomials

the le varieties ii for a complex polynomial f n 1 0 0 with a singular set of complex dimensions at the origin we define a sequence of varieties the le varieties off at 0 the multiplicities of these varieties generalize the milnor number for an isolated singularity in particular we show that ifs n 2 the milnor fibre off is obtained fromb 2n by successively attaching k handles wheren s k n ifs n 1 the milnor fibre off is obtained from a2n manifold with the homotopy type of a bouquet of circles by successively attaching k handles where 2 k n

on boundary values for the laguerre operator in indefinite inner product spaces abstract it is shown that when ly xy 1 x y requires more than one boundary condition at x 0 in order to be self adjoint the more negative the more boundary conditions required

on markov s theorem on zeros of orthogonal polynomials revisited abstract this paper briefly surveys markov s theorem related to zeros of orthogonal polynomials monotonicity of zeros of some families of orthogonal polynomials are reviewed in detail

differential equations having orthogonal polynomial solutions abstract necessary and sufficient conditions for an orthogonal polynomial system ops to satisfy a differential equation with polynomial coefficients of the form l n y i 1 n l i x y i x n y x were found by h l krall here we find new necessary conditions for the equation to have an ops of solutions as well as some other interesting applications in particular we obtain necessary and sufficient conditions for a distribution w x to be an orthogonalizing weight for such an ops and investigate the structure of w x we also show that if the equation has an ops of solutions which is orthogonal relative to a distribution w x then the differential operator ln in must be symmetrizable under certain conditions on w x

answers to some equisingularity questions we give examples of families of hypersurface singularities with constant le numbers constant milnor fibration and non constant topological type answering negatively a question of d massey on the other hand we prove that the constancy of the le numbers implies that the homotopy type of the link is constant along the family as an application we give an example of a flat family of projective reduced and irreducible hypersurfaces having the same homotopy type but different topological type another application is an example of a whitney trivial family of isolated singularities such that the topological type of the projectivised tangent cone is not constant in the family the last example answers negatively a question of o zariski

zeros of orthogonal polynomials generated by canonical perturbations of measures abstract in this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the uvarov perturbation of a positive borel measure d x when the measure is semiclassical then its electrostatic interpretation is given

a general left definite theory for certain self adjoint operators with applications to differential equations we show that any self adjoint operator a bounded or unbounded in a hilbert space h v that is bounded below generates a continuum of hilbert spaces hr r 0 and a continuum of self adjoint operators ar r 0 for reasons originating in the theory of differential operators we call each hr the rth left definite space and each ar the rth left definite operator associated with h a each space hr can be seen as the closure of the domain d ar of the self adjoint operator ar in the topology generated from the inner product arx y x y d ar furthermore each ar is a unique self adjoint restriction of a in hr we show that the spectrum of each ar agrees with the spectrum of a and the domain of each ar is characterized in terms of another left definite space the hilbert space spectral theorem plays a fundamental role in these constructions we apply these results to two examples including the classical laguerre differential expression l in which we explicitly find the left definite spaces and left definite operators associated with a the self adjoint operator generated by l in l2 0 t e t having the laguerre polynomials as eigenfunctions

new representation and factorizations of the higher order ultraspherical type differential equations abstract the paper deals with the class of linear differential equations of any even order 2 4 n 0 which are associated with the so called ultraspherical type polynomials these polynomials form an orthogonal system on the interval 1 1 with respect to the ultraspherical weight function 1 x 2 and additional point masses of equal size at the two endpoints the differential equations of ultraspherical type were developed by r koekoek in 1994 by utilizing special function methods in the present paper a new and completely elementary representation of these higher order differential equations is presented this result is used to deduce the orthogonality relation of the ultraspherical type polynomials directly from the differential equation property moreover we introduce two types of factorizations of the corresponding differential operators of order 2 4 into a product of 2 linear second order operators

orthogonal functions satisfying a second order differential equation let n n 0 be a sequence of functions satisfying a second order differential equation of the form n n n n fn where and fn are smooth functions on the real line r and n is the eigenvalue parameter then we find a necessary and sufficient condition in order for n n 0 to be orthogonal relative to a distribution w and then we give a method to find the distributional orthogonalizing weight w for such an orthogonal function system we also give a necessary and sufficient condition in order that the derived set p n n 0 is orthogonal which is a generalization of lewis and hahn we also give various examples

chain sequences and symmetric generalized orthogonal polynomials in this paper we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials we use this to study the orthogonal polynomials kn m k associated with the probability measure d m k x which is the gegenbauer measure of parameter 1 with two additional mass points at k when k 1 we obtain information on the polynomials kn m which are the symmetric koornwinder polynomials monotonicity properties of the zeros of kn m k in relation to m and k are also given

generalized hahn s theorem abstract let pn x n 0 be an orthogonal polynomial system and l i 0 k a i x d i d d d x a linear differential operator of order k 0 with polynomial coefficients we find necessary and sufficient conditions for a polynomial sequence qn x n 0 defined by q n x l p n r r x n 0 to be also an orthogonal polynomial system we also give a few applications of this result together with the complete analysis of the cases i k 0 1 2 and r 0 and ii k r 1

constructing krall hahn orthogonal polynomials abstract given a sequence of polynomials p n n an algebra of operators a that acts in the linear space of polynomials and an operator d p a with d p p n n p n where n is any arbitrary eigenvalue we construct a new sequence of polynomials q n n by considering a linear combination of m 1 consecutive p n q n p n j 1 m n j p n j using the concept of a d operator we determine the structure of the sequences n j j 1 m such that the polynomials q n n are eigenfunctions of an operator in the algebra a as an application from the classical discrete family of hahn polynomials we construct orthogonal polynomials q n n that are also eigenfunctions of higher order difference operators

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

krall type orthogonal polynomials in several variables for a bilinear form obtained by adding a dirac mass to a positive definite moment functional defined in the linear space of polynomials in several variables explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials

multiplicity of small negative energy solutions for a class of nonlinear schrodinger poisson systems this paper deals with the following nonlinear schrodinger poisson email protected v x u k x f x u h x f x u inr 3 email protected f k x u 2 inr 3 where v x k x and h x are nonnegative continuous functions under appropriate assumptions on v x k x h x and f x u we prove the existence of infinitely many small negative energy solutions by using the variant fountain theorem established by zou recent results from the literature are extended

a method of constructing krall s polynomials we propose a method of constructing orthogonal polynomials pn x krall s polynomials that are eigenfunctions of higher order dierential operators using this method we show that recurrence coecients of krall s polynomials pn x are rational functions of n let p a b m n x be polynomials obtained from the jacobi polynomials p a b n x by the following procedure we add an arbitrary concentrated mass m at the endpoint of the orthogonality interval with respect to the weight function of the ordinary jacobi polynomials we nd necessary conditions for the parameters a b in order for the polynomials p a b m n x to obey a higher order dierential equation the main result of the paper is the following let a be a positive integer and b 1 2 an arbitrary real parameter then the polynomials p a b m n x are krall s polynomials

infinitely many solutions for a nonlinear klein gordon maxwell system abstract in this paper a nonlinear klein gordon maxwell system with solitary wave solution is considered using critical point theory we establish sufficient conditions for the existence of infinitely many solitary wave solutions results obtained complement and improve the existing ones

a pade type approach to non classical orthogonal polynomials abstract by means of pade methods some systems of non classical orthogonal polynomials are introduced it is shown that these polynomials satisfy second order differential equations with polynomial coefficients

infinitely many solutions for fractional kirchhoff schrodinger poisson systems in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild

construction of differential operators having bochner krall orthogonal polynomials as eigenfunctions abstract suppose q n n 0 is a sequence of polynomials orthogonal with respect to the moment functional where is a classical moment functional jacobi laguerre hermite and is a point mass distribution with finite support in this paper we develop a new method for constructing a differential equation having q n n 0 as eigenfunctions

the laguerre type operator in a left definite hilbert space abstract this article completes earlier work concerning the laguerre type differential operator of fourth order set in a weighted sobolev space on 0 it is shown that the operator which is self adjoint in l 2 0 e x r is also self adjoint in the new space whose inner product also involves first and second derivatives further the spectrum remains unchanged the laguerre type polynomials are eigenfunctions and the spectral resolution associated with the operator is still an eigenfunction expansion

existence of ground state solutions for the schrodinger poisson systems abstract in this paper we consider the following schrodinger poisson system e 2 u v x u k x u a x u p 1 u in r 3 e 2 k x u 2 in r 3 with p 1 5 and e 0 we not only investigate the existence of ground state solutions but also employ bifurcation theory to show that the norm of the given ground state solutions tends to zero as the planck constant e 0 moreover we also study some non existence results

existence and non existence of solitary waves for the critical klein gordon equation coupled with maxwell s equations abstract in this paper we investigate existence and non existence of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear klein gordon equation with the maxwell equations when the nonlinearity exhibits critical growth we prove a non existence result by a poho aev type argument and then adding a suitable perturbation we recover existence of at least a radially symmetric solution we overcome the lack of compactness relying on the brezis nirenberg method

left definite theory with applications to orthogonal polynomials in the past several years there has been considerable progress made on a general left definite theory associated with a self adjoint operator a that is bounded below in a hilbert space h the term left definite has its origins in differential equations but littlejohn and wellman l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 generalized the main ideas to a general abstract setting in particular it is known that such an operator a generates a continuum h r r 0 of hilbert spaces and a continuum of a r r 0 of self adjoint operators in this paper we review the main theoretical results in l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 moreover we apply these results to several specific examples including the classical orthogonal polynomials of laguerre hermite and jacobi

djkm algebras and non classical orthogonal polynomials abstract we describe families of polynomials arising in the study of the universal central extensions of lie algebras introduced by date jimbo kashiwara and miwa 1983 6 in their work on the landau lifshitz equations we show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

on a generalization of bessel functions satisfying higher order differential equations bessel type functions j m x 0 with two parameters 12 and m 0 which include the classical normalized bessel function for m 0 are introduced as a certain confluent limit of koornwinder s laguerre type polynomials l nn x n n0 for any n0 they arise as solutions of a spectral dependent differential equation of order 2 4 a necessary and sufficient condition to transform the differential equation into symmetric form is given in terms of an overdetermined system of linear equations it is shown that for 0 1 2 a solution of this problem exists and leads to a symmetric fourth sixth and eighth order differential equation respectively for any 12 we also derive a second order differential equation but with coefficients depending nonlinearly on the eigenvalue parameter finally two differential expressions of order 2 2 n0 are constructed which map the classical bessel functions onto their nonclassical counterparts and vice versa this result may be used to establish an orthogonality relation for the bessel type functions in a distributional sense with respect to the distributional weight function 2 1 x2 1 m x denoting the point mass centered at the origin

topological equisingularity of hypersurfaces with 1 dimensional critical set abstract we focus on topological equisingularity of families of holomorphic function germs with 1 dimensional critical set we introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular we show that if a family of germs with 1 dimensional critical set has constant generic le numbers then it is equisingular at the critical set and hence topologically equisingular answering a question of d massey 13 it is worth to remark that this does not happen for higher dimensional critical set 5 we use these topological triviality results to modify the definition of singularity stem present in the literature introducing and characterising topological stems being this concept closely related with arnold s series of singularities we provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non flat deformation with singular special fibre and smooth generic fibre

standing waves for quasilinear schrodinger equations with indefinite potentials abstract we consider quasilinear schrodinger equations in r n of the form u v x u u u 2 g u where g u is 4 superlinear unlike all known results in the literature the schrodinger operator v is allowed to be indefinite hence the variational functional does not satisfy the mountain pass geometry by a local linking argument and morse theory we obtain a nontrivial solution for the problem in case that g is odd we get an unbounded sequence of solutions

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

sobolev orthogonal polynomials and spectral differential equations we find necessary and sufficient conditions for a spectral differential equation n ln y x 52ti x yw x kny x 1 to have sobolev orthogonal polynomials of solutions which are orthogonal rel ative to the sobolev pseudo inner product c t p q pqdn i p q du

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

semiclassical solutions for the nonlinear schrodinger maxwell equations abstract in this paper by using variational methods and critical point theory we study the existence of semiclassical solutions for the following nonlinear schrodinger maxwell equations e 2 u v x u u f x u in r 3 4 u 2 in r 3 where e 0 and v x 0 for all x r 3 under some weaker assumptions on v and f we prove that the system has at least one nontrivial solution for sufficient small e 0

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials