connections between kelvin functions and zeta functions with applications a very simple method for resuming infinite series of the type sup approximately l 1 or 1 l 1 l n 1 2 cosh la k n 1 2 lb is presented as a result a series expansion in powers of a and b is obtained relevant to many physical applications

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

generalized multidimensional epstein zeta functions explicit formulas are given for the expression of multiple sum zeta functions of the type l1 ln a1 l1 c1 2 an ln cn 2 m2 s where ai 0 m2 0 i 1 n in terms of riemann and hurwitz zeta functions and in terms of kelvin functions using different analytical continuation methods several formulas especially useful for different values of the involved parameters are obtained

multiplicity of stationary solutions to the euler poisson equations abstract consider the system of euler poisson as a model for the time evolution of gaseous stars through the self induced gravitational force we study the existence uniqueness and multiplicity of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy a priori these results generalize the previous works on the irrotational or the rotational gaseous stars around an axis and then they hold in more general physical settings under the assumption of radial symmetry the monotonicity properties of the radius of the gas with respect to either the strength of the velocity field or the center density are also given which yield the uniqueness under some circumstances

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

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

uniqueness of positive solutions of a quasilinear dirichlet problem with exponential nonlinearity uniqueness of positive solutions of the following equation has been obtained where b n is the ball of unit radius and 0 moreover if n 2 then the solutions are nondegenerate

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

some amalgams in characteristic 3 related toco1 abstract co 1 has parabolic subgroups of the shape 3 6 2 m 12 and 3 1 4 sp 4 3 2 lyons simple group ly has parabolics of the form 3 5 m 11 2 and 3 2 4 a 5 d 8 we will characterize these parabolics and similar ones found in subgroups of co 1 using the amalgam method

zeta functions on the non positive real axis physical applications of zeta function regularisation require explicit knowledge of the relevant zeta function somewhere on the non positive real axis the author presents a simple and very general method for evaluating zeta functions z s sigma m lambda m s at certain discrete points s np or 0 n 0 1 2 on this axis where p is defined by the condition that lambda mp be a polynomial in all summation indices z np is shown to be a polynomial in the various parameters on which the eigenvalues lambda m depend the polynomials z np generalise the bernoulli polynomials and possess interesting properties these statements apply to multidimensional e g epstein and other zeta functions as well as to single sum zeta functions physical applications in quantum field theory are briefly indicated

positive radial solutions for quasilinear systems in an annulus abstract we show that either superlinearity or sublinearity assumptions can guarantee the existence of positive radial solutions for quasilinear systems involving the p laplacian

orbital stability investigations for travelling waves in a nonlinearly supported beam abstract we consider the fourth order wave equation t t x x x x f 0 x t r r with a nonlinearity f vanishing at 0 travelling waves x t u x c t satisfy the ode u c 2 u f u 0 on r and for the case f u e u 1 the existence of at least 36 solitary travelling waves was proved in 4 by computer assisted means we investigate the orbital stability of these solutions via computation of their morse indices and using results from 11 and 12 in order to achieve our results we make use of both analytical and computer assisted techniques

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

morse index and uniqueness for positive solutions of radial p laplace equations we study the positive radial solutions of the dirichlet problem p u f u 0 in b u 0 in b u 0 on b where p u div u p 2 u p 1 is the p laplace operator b is the unit ball in r n centered at the origin and f is a c 1 function we are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the morse index in particular we show that positive radial solutions of mountain pass type have morse index 1 in the subspace of radial functions of w 1 p 0 b we use this to prove uniqueness and nondegeneracy of positive radial solutions when f is of the type u s u q and p 2

uniqueness of positive radial solutions for δu u up 0 on an annulus abstract we prove uniqueness of positive radial solutions to the semilinear elliptic equation u u u p 0 p 1 subject to the dirichlet boundary condition on an annulus in r n n 3 as a by product our argument also provides a much simpler if not the simplest new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space r n

on the zeta function regularization of a two dimensional series of epstein hurwitz type as a further step in the general program of zeta function regularization of multiseries expressions some original formulas are provided for the analytic continuation to any value of s of two dimensional series of epstein hurwitz type namely n1 n2 0 a1 n1 c1 2 a2 n2 c2 2 s where the aj are positive reals and the cj are not simultaneously nonpositive integers they come out from a generalization to hurwitz functions of the zeta function regularization theorem of the author and romeo phys rev d 40 436 1989 for ordinary zeta functions for s k 0 2 with k 1 2 3 the final results are in fact expressed in terms of hurwitz zeta functions only for general s they also involve bessel functions a partial numerical investigation of the different terms of the exact algebraic equations is also carried out as a by product the series n 0exp a n c 2 a c 0 is conveniently calculated in terms of them

multiple solutions for a semilinear boundary value problem a computational multiplicity proof abstract we prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second order elliptic boundary value problems with a nonlinearity which grows superlinearly at the proof makes massive use of computer assistance after approximate solutions have been computed by a numerical mountain pass algorithm combined with a newton iteration to improve accuracy a fixed point argument is used to show the existence of exact solutions close to the approximations

uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations in this article we are concerned with the semilinear elliptic equation where n 2 p 1 and k lxl 0 in r n the correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived in addition we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on k and p furthermore based on the pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions we obtain the whole structure of radial solutions depending on various situations

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

uniqueness and profile of positive solutions of a critical exponent problem with hardy potential abstract we obtain the uniqueness and blow up profile of positive least energy solution of a critical exponent problem with hardy potential in a ball

an elementary proof of the uniqueness of positive radial solutions of a quasilinear dirichlet problem we consider the quasilinear elliptic equation text div left nabla u m 2 nabla u right u p lambda u m 2 u in in b where b is a ball or an annulus in n 1 m n p is a positive real number and e using a generalized poho aev type variational identity of ni serrin or pucci and serrin and an elementary calculus lemma we establish uniqueness of positive radial solutions for the dirichlet boundary condition if either left text i right b text is a ball lambda underset raise0 3em hbox smash scriptscriptstyle geqslant text 0 1 p underset raise0 3em hbox smash scriptscriptstyle leqslant frac mn n m text for m n text and 1 p infty text for m n text or left text ii right b text is an annulus lambda in mathbb r text and p frac mn n m

sharp numerical inclusion of the best constant for embedding h 0 1 ź ź l p ź on bounded convex domain in this paper we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding h 0 1 l p on a bounded convex domain in r 2 we estimate the best constant by computing the corresponding extremal function using a verified numerical computation verified numerical inclusions of the best constant on a square domain are presented

a uniqueness result for a nonlocal equation of kirchhoff type and some related open problem abstract we study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of kirchhoff type a b u 2 d x u f u in u 0 here is a bounded open set in r n with smooth boundary a b are positive real numbers and f r r is a continuous function in particular we give an answer to an open problem recently proposed by b ricceri

a computer assisted study of uniqueness of ground state solutions for the problem here u u x du u p au q bu r 0 x r n lim x u x 0 with constants 1 p0 uniqueness of radial solution calledground state solution is not known we present a procedure which opens the way to produce computer assisted proofs of uniqueness for specific p q r and n

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

new exact multiplicity results with an application to a population model we obtain some new exact multiplicity results for the dirichlet boundary value problem on a unit ball b n in r n we consider several classes of nonlinearities f u including both positive and sign changing cases a crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem as an application we obtain exact multiplicity results for the holling tanner population model

a uniqueness result for a semilinear elliptic problem a computer assisted proof abstract starting with the famous article a gidas w m ni l nirenberg symmetry and related properties via the maximum principle comm math phys 68 1979 209 243 many papers have been devoted to the uniqueness question for positive solutions of u u u p in u 0 on where p 1 and ranges between 0 and the first dirichlet eigenvalue 1 of for the case when is a ball uniqueness could be proved mainly by ode techniques but very little is known when is not a ball and then only for 0 in this article we prove uniqueness for all 0 1 in the case 0 1 2 and p 2 this constitutes the first positive answer to the uniqueness question in a domain different from a ball our proof makes heavy use of computer assistance we compute a branch of approximate solutions and prove existence of a true solution branch close to it using fixed point techniques by eigenvalue enclosure methods and an additional analytical argument for close to 1 we deduce the non degeneracy of all solutions along this branch whence uniqueness follows from the known bifurcation structure of the problem

solutions of euler poisson equations for gaseous stars in this paper we study the euler poisson equations governing gas motion under self gravitational force we are interested in the evolution of the gaseous stars for which the density function has compact support we establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary the boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in l ly for both euler equations with damping and the euler poisson equations existence non existence uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function and the phenomena of the blowup that is the collapsing of the star to a single point with finite mass as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations

existence results of steady states of semilinear reaction diffusion equations and their applications partially supported by the applied mathematical sciences subprogram of the offlice of energy research u s department of energy under contract w 31 109 eng 38 while a visit ing scientist at argonne national laboratory during the summer of 1987 and m part by the board of regents of the state of louisiana under grant leqsf 86 89 rd a 11 partially supported by the applied mathematical sciences subprogram of the office of energy research u s department of energy under contract w 31 109 eng 38 this author thanks the department of mathematics university of southwestern louisiana for an invitation to spend a semester there in the fall of 1986 during which the research reported in this paper was started

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

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

uniqueness of positive solutions of a n laplace equation in a ball in rn with exponential nonlinearity abstract let n 2 and r n be a bounded domain then by trudinger moser embedding w 0 1 n is embedded in an orlicz space consisting of exponential functions consider the corresponding semilinear n laplace equation with critical or sub critical exponential nonlinearity in a ball b r with dirichlet boundary condition in this paper we prove that under suitable growth conditions on the nonlinearity there exists an 0 0 and a corresponding r 0 0 0 such that for all 0 r r 0 the problem admits a unique nondegenerate positive radial solution u with u 0

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

multispike solutions for the brezis nirenberg problem in dimension three abstract we consider the problem u u u 5 0 u 0 in a smooth bounded domain in r 3 under zero dirichlet boundary conditions we obtain solutions to this problem exhibiting multiple bubbling behavior at k different points of the domain as tends to a special positive value 0 which we characterize in terms of the green function of

a note on a nonlinear elliptic problem with a nonlocal coefficient abstract in this paper we investigate a nonlocal and nonlinear elliptic problem p a u 2 d x u u u p in u 0 on where n 3 r n is a bounded domain with smooth boundary a is a nondegenerate continuous function p 1 and r we show several effects of the nonlocal coefficient a on the structure of the solution set of p we first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem this allows us to get unbounded continua of solutions u of p a rich variety of new bifurcation and multiplicity results are observed we also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way lastly we give some remarks from the variational point of view

pushing up point stabilizers ii abstract this paper is a continuation of the paper g parmeggiani pushing up point stabilizers i j algebra 319 9 2008 3854 3884 under the same hypotheses we determine those amalgams which involve sl n q n 3 as factors

multiple harmonic oscillator zeta functions the partition function for a quantum mechanical system can at least formally be expressed as a high temperature series in terms of the zeta function of the system s hamiltonian this is done explicitly for a system of non interacting harmonic oscillators the hamilton zeta function belongs to a class of zeta functions sigma f ni n1 omega 1 n2 omega 2 nn omega n a s summed over non negative integers ni very little is known about these zeta functions and an investigation of some of their properties is presented

baumann components of finite groups of characteristic p reduction theorems abstract we continue the project started in 8 to describe the structure of the finite groups g of characteristic p in terms of their baumann components and the conjugacy class bau p g the reduction theorem proved in 8 allows to assume that g has a unique baumann component in this paper we use this property to determine the isomorphism type of g o p g and the action of g on 1 z o p g in addition we prove reduction theorems which allow to focus on groups g which satisfy g o p g sl n q sp 2 n q or g 2 q and o p g b for b bau p g

nontrivial solutions for the equations of monge ampere type on etudie les solutions strictement convexes des equations de type monge ampere detu xd i xd j u p dans la boule b x 0 p n est un parametre reel on demontre l existence et l unicite

some superlinear problems with nonlocal diffusion coefficient abstract we study a superlinear elliptic problem with a non local diffusion coefficient we show that there exists a drastic change on the structure of the set of positive solutions when the non local coefficient grows fast enough to infinity we combine mainly sub super and bifurcation methods to obtain our results

a computer assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam abstract for a nonlinear beam equation with exponential nonlinearity we prove existence of at least 36 travelling wave solutions for the specific wave speed c 1 3 this complements the result in smets van den berg homoclinic solutions for swift hohenberg and suspension bridge type equations j differential equations 184 2002 78 96 stating that for almost all c 0 2 there exists at least one solution our proof makes heavy use of computer assistance starting from numerical approximations we use a fixed point argument to prove existence of solutions close to the computed approximations

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

new solutions of the gelfand problem abstract we propose a computer assisted method for proving existence and obtaining enclosures of solutions of nonlinear elliptic boundary value problems which may be successful in cases where purely analytic methods are not this method is an extension of our previous work interval comput 3 1994 106 121 z angew math mech 76 1996 171 176 on the gelfand problem to nonconvex domains in r 2 here we are able to prove the existence of a solution branch which was unknown so far