master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

integrable three dimensional lattices following the work of zakharov and shabat 1974 the authors derive nonlinear equations which are integrable through a discrete gel fand levitan integral equation in two resp one resp zero continuous and one resp two resp three discrete variables

multiple positive solutions for quasilinear problems with indefinite sublinear nonlinearity abstract we have established multiplicity of nontrivial solutions for the quasilinear elliptic problem p u h x u 1 g x u in u 0 in u 0 on where r n is a smooth bounded domain p 1 1 p h is a measurable function and g r r is a caratheodory function such that g x 0 0 which is asymptotically linear we also prove existence and nonexistence results when g x t k x t p 1

new integrable systems related to the relativistic toda lattice the bi hamiltonian structure of the relativistic toda lattice is exploited to introduce some new integrable lattice systems their integrable discretizations are obtained by means of the general procedure proposed recently by the author backlund transformations between the new systems and the relativistic toda lattice in both the continuous and discrete time formulations are established

critical groups of pairs of critical points produced by linking subsets courant institute of mathematical sciences new york university 251 mercer street new york new york 10012received december 11 1996contents1 introduction 2 linking 2 1 definition and examples 2 2 two abstract lemmas 3 critical point theorems 3 1 functions on hilbert riemannian manifolds 3 2 functions on hilbert spaces 3 3 functions on manifolds with boundary 4 applications to pdes and odes 4 1 an elliptic boundary value problem witha concave nonlinearity 4 2 nonconstant time periodic solutions of a system ofodes 4 3 an alternative for a superlinear problem

rational solutions of a differential difference kdv equation the toda equation and the discrete kdv equation in this paper a series of rational solutions are presented for a differential difference analogue of the kdv equation the toda equation and the discrete kdv equation these rational solutions are obtained using hirota s bilinear formalism and backlund transformations the crucial step is the use of nonlinear superposition formulae

multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin abstract in this paper we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin the solutions are obtained from the versions of mountain pass lemma and linking theorem

multiplicity results for some elliptic problems with concave nonlinearities implies that the variational functionalassociated with 1 is bounded below and coercive coercive ellipticproblems of this type with 0 have been studied by ambrosetti andmancini 3 struwe 9 ambrosetti and lupo 2 and other authors here we prove the existence of more solutions for 0 sufficiently small

computation of densities and fluxes of nonlinear differential difference equations direct methods to find conserved densities and fluxes of differentialdifference equations are presented and illustrated for the kacvan moerbeke kvm and modified volterra lattices a miura map whi

multiple solutions for a semilinear problem with combined terms and nonlinear boundary conditions abstract we consider the problem u u f x u in u h x u q 2 u on where r n is a smooth bounded domain n 3 1 q 2 and h belongs to an appropriated lebesgue space in our main results we suppose that f is an asymptotically linear function and we obtain multiplicity of solutions when the norm of h is small we also present a multiplicity result in the case that f is nonquadratic at infinity

comatrix corings galois corings descent theory and a structure theorem for cosemisimple corings we study the corings whose category of right comodules has a finitely generated projective generator in order to extricate the structure of these corings we introduce the notion of comatrix coring comatrix corings allow as well to give a complete description of all cosemisimple corings

multiple solutions for some elliptic equations with a nonlinearity concave at the origin in this paper we establish the existence of multiple solutions for the semilinear elliptic problem u u q 2 u au g u in u 0o n where r n is a bounded domain with smooth boundary g r r is a function of class c 1 such that g 0 g 0 0 0i s ar eal parameter a r a nd 1 q 2

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

multiple solutions to p laplacian problems with concave nonlinearities in this paper we study the p laplacian type elliptic problems with concave nonlinearities using some asymptotic behavior of f at zero and infinity three nontrivial solutions are established

semilinear elliptic problems with asymmetric nonlinearities abstract in this paper we are concerned on the semilinear elliptic problem u u q 2 u a u b u p 1 in u 0 on where r n is a bounded domain with regular boundary 1 q 2 p 2 if a is between two eigenvalues we get the existence of three nontrivial solutions for the problem above

solutions to semilinear elliptic problems with combined nonlinearities abstract we are concerned with the following nonlinear dirichlet problem u h x u q f x u 0 u h 0 1 0 where is a bounded smooth domain in rn n 1 and h x l f x s is asymptotically linear with respect to s at infinity by a variant version of mountain pass theorem we prove that there exist at least two nonnegative solutions under suitable assumptions on f x s our methods also work for the cases where f x s is superlinear or linear in s

on comonadicity of the extension of scalars functors abstract building on a categorical approach of g janelidze and w tholen to descent theory for modules we show how this theory can be presented at the level of enriched categories specializing to the monoidal category of bimodules over a separable algebra this gives a criterion for comonadicity of the extension of scalars functor associated to an extension of not necessarily commutative rings as an application of this criterion some known results on the comonadicity of such functors are obtained

elliptic problems on complete non compact riemannian manifolds with asymptotically non negative ricci curvature abstract in this paper we discuss the existence and non existence of weak solutions to parametric equations involving the laplace beltrami operator g in a complete non compact d dimensional d 3 riemannian manifold m g with asymptotically non negative ricci curvature and intrinsic metric d g namely our simple model is the following problem g w v w f w in m w 0 in m where v is a positive coercive potential is a positive bounded function is a real parameter and f is a suitable continuous nonlinear term the existence of at least two non trivial bounded weak solutions is established for large value of the parameter requiring that the nonlinear term f is non trivial continuous superlinear at zero and sublinear at infinity our approach is based on variational methods no assumptions on the sectional curvature as well as symmetry theoretical arguments are requested in our approach

existence and multiplicity results for a class of semilinear elliptic equations abstract we study the existence and multiplicity of nonnegative solutions as well as the behavior of corresponding parameter dependent branches to the equation u 1 u u m u n in a bounded domain r n endowed with the zero dirichlet boundary data where 0 m 1 and n 0 when 0 the obtained solutions can be seen as steady states of the corresponding reaction diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination in addition to the main new results we formulate a few relevant conjectures

multiplicity of solutions for a quasilinear elliptic problem via the cohomological index abstract in this paper we study a quasilinear elliptic problem with the dirichlet boundary condition in a bounded domain involving the operator a u p u q u assuming that the nonlinearity has a concave convex behavior we obtain some multiplicity results more precisely we obtain five nontrivial solutions two by a minimization argument two by the mountain pass theorem and the other by a cohomological linking theorem

sobolev versus holder local minimizers in degenerate kirchhoff type problems abstract in this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of kirchhoff type due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a priori estimates and sobolev versus holder local minimizers type of results we prove that results similar to these hold true or not depending on how degenerate the problem is we apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations considering several different interactions between the nonlocal term and the nonlinearity

the inhomogeneous toda lattice its hierarchy and darboux backlund transformations the authors show how one can construct hierarchies of nonlinear differential difference equations with n dependent coefficients among these equations they present explicitly a set of inhomogeneous toda lattice equations which are associated with a discrete schrodinger spectral problem whose potentials diverge asymptotically then they derive a new darboux transformation which allows them to get bounded solutions for the equations presented before and apply it in a specially simple case when the solution turns out to be expressed in terms of hermite polynomials

non uniformly asymptotically linear p laplacian problems abstract we relax conditions on uniform convergences of f x s s p 2 s which are essential in many results on multiplicity for asymptotically linear p laplacian problems

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

evolution equations associated with the discrete analog of the matrix schrodinger spectral problem solvable by the inverse spectral transform through the generalized wronskian technique we derive the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger spectral problem for such equations we briefly discuss soliton solutions continuum limit and backlund transformations

galois theory for comatrix corings descent theory morita theory frobenius and separability properties el kaoutit and gomez torrecillas introduced comatrix corings generalizing sweedler s canonical coring and proved a new version of the faith fully flat descent theorem they also introduced galois corings as corings isomorphic to a comatrix coring in this paper we further investigate this theory we prove a new version of the joyal tierney descent theorem and generalize the galois coring structure theorem we associate a morita con text to a coring with a fixed comodule and relate it to galois type properties of the coring an affineness criterion is proved in the situation where the cor ing is coseparable further properties of the morita context are studied in the situation where the coring is co frobenius

computation of conservation laws for nonlinear lattices an algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented the algorithm is implemented in mathematica and can be used as an automated integrability test with the code diffdens m conserved densities are obtained for several well known lattice equations for systems with parameters the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist

an approach to directly construct exact solutions of nonlinear differential difference equations abstract in this paper a constructive method for exactly solving nonlinear differential difference equations nddes is presented the ndde which includes hybrid lattice discretized mkdv lattice and modified volterra lattice is chosen to illustrate this approach some new solutions of these lattices are obtained