interval criteria for oscillation of nonlinear second order dynamic equations on time scales interval oscillation criteria are established for a second order nonlinear dynamic equation on time scales by utilizing a generalized riccati technique and an integral averaging technique the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives

intertwining operators into dolbeault cohomology representations let gl be the quotient of a semisimple lie group g by the centralizer l of a torus the space of dolbeault cohomology sections of a holomorphic line bundle over gl is a natural place to realize interesting irreducible unitary representations of g and was first studied for this purpose by bott and schmid zuckerman and vogan later introduced derived functor modules to provide an algebraic analog of these representations the authors give a nonzero integral intertwining operator from derived functor modules realized in the langlands classification to the dolbeault cohomology representations under the assumption that l and g have the same real rank

holder bounds and regularity of emerging free boundaries for strongly competing schrodinger equations with nontrivial grouping abstract we study interior regularity issues for systems of elliptic equations of the type u i f i x j i a i j u i u i p 1 u j p 1 set in domains r n for n 1 the paper is devoted to the derivation of c 0 estimates that are uniform in the competition parameter 0 as well as to the regularity of the limiting free boundary problem obtained for the main novelty of the problem under consideration resides in the non trivial grouping of the densities in particular we assume that the interaction parameters a i j are only non negative and thus may vanish for specific couples i j as a main consequence in the limit densities do not segregate pairwise in general but are grouped in classes which in turn form a mutually disjoint partition moreover with respect to the literature we consider more general forcing terms sign changing solutions and an arbitrary p 0 in addition we present a regularity theory of the emerging free boundary defined by the interface among different segregated groups these equations are very common in the study of bose einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

dolbeault cohomological realization of zuckerman modules associated with finite rank representations abstract the dolbeault complex of an arbitrary finite dimensional homogeneous holomorphic vector bundle on certain open orbits in a generalized flag manifold is studied the complex is shown to have the closed range property and the resulting cohomologies form the maximal globalization representations of the corresponding zuckerman modules

on a critical schrodinger system in r4 with steep potential wells abstract in this paper we study the following two component elliptic system u a x a 0 u u 3 v 2 u 0 in r 4 v b x b 0 v v 3 u 2 v 0 in r 4 u v h 1 r 4 h 1 r 4 where a 0 b 0 r are constants 0 and r are parameters a x b x 0 are potentials by using variational methods we prove the existence and nonexistence of general ground state solutions maybe semi trivial and ground state solutions of the above system under some further conditions on the potentials a x b x and the parameters it is worth pointing out that the cubic nonlinearities and the coupled terms of the above system are all of critical growth owing to the sobolev embedding theorem furthermore by introducing some ideas which are different from that in the literature the phenomenon of phase separation of ground state solutions of the above system is also obtained without any symmetry conditions

uniform holder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames the main result of this paper is a general holder estimate in a class of singularly perturbed elliptic systems this estimate is applied to the study of the well known burke schuman approximation in flame theory after reviewing some classical cases equidiffusional case high activation energy approximation we turn to the non equidiffusional case and to nonlinear diffusion models the limiting problems are nonlinear elliptic equations they have holder or lipschitz maximal global regularity a natural question is then whether this regularity is kept uniformly throughout the approximation process and we show that this is true in general

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

second order oscillation of forced functional differential equations with oscillatory potentials new oscillation criteria are established for second order differential equations containing both delay and advanced arguments of the form k t x t p t x t t a 1x t t q t x s t b 1x s t e t t 0 where a 1 and b 1 k p q e t s are continuous real valued functions k t 0 is nondecreasing t and o are nondecreasing t t t s t t and lim t t t the potentials p q and e are allowed to change sign and the information on the whole half line is not required as opposed to the usual case in most articles among others as an application of the results we are able to deduce that every solution of x t m 1sintx t p12 m 2costx t p6 cos2t m 1 m 2 0 is oscillatory provided that either m 1 orm 2 is sufficiently large

some comparison and oscillation results for first order differential equations and inequalities with a deviating argument abstract in this paper we are dealing with first order differential equations and inequalities with a deviating argument and we give some new necessary sufficient and necessary and sufficient conditions concerning a the comparison of oscillatory and asymptotic behavior of solutions of 1 first order vector differential equations and corresponding to them operator and scalar differential equations with a deviating argument and 2 first order differential equations and corresponding to them inequalities with a deviating argument and b the existence of a certain type of nonoscillatory solutions of differential equations and inequalities with a deviating argument what is important in our opinion is the fact that some of the obtained results concern not only the cases of constant sign coefficients but also the cases where the coefficients of equations and inequalities under consideration are of non constant sign

oscillation and nonoscillation of advanced differential equations with variable coefficients the oscillation and nonoscillation of the advanced differential equations x t p t x t 0 t t0 and x t i 1npi t x t i 0 t t0 are investigated where p t pi t c t0 0 and i are positive constants at first a sharp sufficient condition for the oscillation of eq is obtained then the result is generalized to eq these results improve the corresponding conclusions derived by ladas and stavroulakis j differential equations 44 1982 134 152 next two examples are given to illustrate the advantages of our results finally the sufficient conditions for these two equations to be nonoscillatory are also obtained

boundary value maps szego maps and intertwining operators we consider one series of unitarizable representations the cohomological induced modules aq a with dominant regular infinitesimal character the minimal k type ir v of aq a determines a homogeneous vector bundle v g k the derived functor modules can be realized on the solution space of a first order differential operator da on v barchini knapp and zierau gave an explicit integral map s from the derived functor module realized in the langlands classification into the space of smooth sections of the vector bundle v g k in this paper we study the asymptotic behavior of elements in the image of s we obtain a factorization of the standard intertwining opeartors into the composition of the szego map s and a passage to boundary values 0 introduction let g be a linear connected semisimple lie group and let k be a maximal compact subgroup of g we assume that rank g rank k in this situation the cohomologically induced representation aq a with dominant regular infinitesimal character can be realized on solution spaces of invariant differential equations in g k see wo b 2 the minimal k type r v of aq a determines an homogeneous vector bundle vtr g k corresponding to certain choices of positive roots there is defined a certain homogeneous first order differential operator d on vtu in bkz and b 1 the authors gave an explicit nonzero integral map s from the derived functor module realized in the langlands classification to the space of smooth sections of the vector bundle vt g k the image of s is annihilated by the differential operator d when the representation under consideration is in the discrete series of g the differential operator vd coincides with schmid s differential operator s in this case the s transform was introduced by knapp and wallach in kw the realization of irreducible unitary representations in the g k picture generalizes bargmann s formulation of holomorphic discrete series of su 1 1 bar in terms of semisimple lie groups functions on the circle suggest non unitary principal series and holomorphic functions on the disk suggest discrete series representations the s transform in kw reduces in limiting cases limits of discrete series for su 1 1 to the szeg6 kernel for the disk see k 1 thus we call s a szeg6 map received by the editors february 17 1995 and in revised form october 9 1995 1991 mathematics subject classification primary 22c05 22e45 22e46 1997 american mathematical society 1877 this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 42 59 utc all use subject to http about jstor org terms

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

sign changing semi classical solutions of the brezis nirenberg problems with jump nonlinearities in high dimensions abstract in this paper we consider the following elliptic equation e 2 u u u u p 2 u u 2 2 u in u 0 on where r n n 4 is a bounded domain u max u 0 0 are constants e 0 is a small parameter and 2 p 2 2 n n 2 we obtain a least energy sign changing solution to this problem for e 0 that is sufficiently small by means of the variational method furthermore by combining the elliptic and local energy estimates we study the concentration behavior of this least energy sign changing solution and the location of the positive and negative spikes as e 0 our results partially complete the studies in 6 2011 in the sense that due to sobolev embedding only subcritical cases are considered in that paper

on critical systems involving fractional laplacian abstract consider the following non local critical system 0 1 s u 1 u 1 u 2 2 u 2 u 2 u v in s v 2 v 2 v 2 2 v 2 u v 2 v in u 0 v 0 in r n where s is fractional laplacian 0 s 1 and all 1 2 1 2 0 2 2 n n 2 s is a fractional sobolev critical exponent n 2 s 1 2 and is an open bounded domain in r n with lipschitz boundary under proper conditions we establish the existence result of the ground state solution to system 0 1

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

existence and symmetry of positive ground states for a doubly critical schrodinger system n n 2 and 1 1 satisfying 2 this problem is related to coupled nonlinear schrodinger equations with critical exponent for bose einstein condensate for different ranges of n and 0 we obtain positive ground state solutions via some quite different methods which are all radially symmetric it turns out that the least energy level depends heavily on the relations among and 2 besides for sufficiently small 0 positive solutions are also obtained via a variational perturbation approach note that the palais smale condition can not hold for any positive energy level which makes the study via variational methods rather complicated

multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent abstract this paper deals with the following system linearly coupled by nonlinear elliptic equations u 1 u u 2 2 u v x v 2 v v 2 2 v u x u v 0 on here is a smooth bounded domain in r n n 3 1 2 1 are constants 1 is the first eigenvalue of h 0 1 2 2 n n 2 is the sobolev critical exponent and r is a coupling parameter by variational method we prove that this system has a positive ground state solution for some 0 via a perturbation argument we show that this system also admits a positive higher energy solution when is small moreover the asymptotic behaviors of the positive ground state and higher energy solutions as 0 are analyzed

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two abstract we prove using variational methods the existence in dimension two of positive vector ground states solutions for the bose einstein type systems u 1 u 1 u e u 2 1 v e u v 1 in v 2 v 2 v e v 2 1 u e u v 1 in u v h 0 1 where is a bounded smooth domain 1 2 1 the first eigenvalue of h 0 1 1 2 0 and is either positive small or large or negative small the nonlinear interaction between two bose fluids is assumed to be of critical exponential type in the sense of j moser for small solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power like nonlinearities

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

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

standing waves of a weakly coupled schrodinger system with distinct potential functions abstract the standing wave solutions of a weakly coupled nonlinear schrodinger system with distinct trapping potential functions in r n 1 n 3 are considered this type of system arises from models in bose einstein condensates theory and nonlinear optics the existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value which is explicitly defined in terms of potential functions and system parameters it is also shown that such solutions concentrate near the minimum points of potential functions and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition variational approach is used for the existence and concentration of positive solutions

multiple bound states of nonlinear schrodinger systems this paper is concerned with existence of bound states for schrodinger systems which have appeared as several models from mathematical physics we establish multiplicity results of bound states for both small and large interactions this is done by different approaches depending upon the sizes of the interaction parameters in the systems for small interactions we give a new approach to deal with multiple bound states the novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows for large interactions we use a minimax procedure to distinguish solutions by analyzing their morse indices

positive ground state solutions for an elliptic system with hardy sobolev critical exponent growth this paper considers a nonlinear elliptic system with hardy sobolev critical exponent growth several existence results on the positive ground state solutions to the system are established under proper conditions this paper considers a nonlinear elliptic system with hardy sobolev critical exponent growth several existence results on the positive ground state solutions to the system are established under proper conditions

positive least energy solutions for a coupled schrodinger system with critical exponent abstract in this paper we consider the following coupled schrodinger system with doubly critical exponents which can be seen as a counterpart of the brezis nirenberg problem u 1 u 1 u 5 u 2 v 3 x v 2 v 2 v 5 v 2 u 3 x u 0 v 0 x u v 0 x where r 3 is a smooth bounded domain 1 2 0 1 2 0 and 0 under certain conditions on 1 2 and we show that this problem has at least one positive least energy solution

multiple sign changing and semi nodal solutions for coupled schrodinger equations we study the following coupled schrodinger equations which have appeared as several models from mathematical physics u1 1u1 1u13 u1u22 x u2 2u2 2u23 u12u2 x u1 u2 0on here rn n 2 3 is a smooth bounded domain 1 2 1 2 are all positive constants we show that for each k n there exists k 0 such that this system has at least k sign changing solutions i e both two components change sign and k semi nodal solutions i e one component changes sign and the other one is positive for each fixed 0 k

on nonoscillation of mixed advanced delay differential equations with positive and negative coefficients for a mixed advanced delay differential equation with variable delays and coefficients email protected t a t x g t b t x h t 0 t t 0 where a t 0 b t 0 g t t h t t explicit nonoscillation conditions are obtained

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

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

positive ground state solutions for a nonlinearly coupled schrodinger system with critical exponents in r4 abstract consider the following nonlinear schrodinger equations u 1 u 1 u 3 3 u 2 v u v 2 v 3 x v 2 v 2 v 3 3 u v 2 u 2 v u 3 x u 0 v 0 in u v 0 on where r 4 is a smooth bounded domain 1 2 0 1 2 0 0 and is the first eigenvalue of with the dirichlet boundary condition note that all the cubic nonlinearities including the coupling terms are of critical growth the existence of positive ground state solutions for the system is established

positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems in this study we consider the following coupled elliptic system with a sobolev critical exponent p u1 1u1 1u1p1 1 1u12 1 u12 2 1u22 2 x u2 2u2 2u2p2 1 2u22 1 u12 2u22 2 1 x u1 u2 0in u1 u2 0on where rn is a bounded smooth domain n 5 2 0 for j 1 2 and 1 is the first eigenvalue of with the dirichlet boundary condition we demonstrate the existence of a positive ground state solution for problem p when the coupling parameter 1 2 under some other conditions we show the nonexistence of positive solutions for p when n 3 we also construct multiple nontrivial solutions and sign changing solutions for the following system u1 1u1 1u13 u22u1 x u2 2u2 2u23 u12u2 x u1 u2 0on where rn is a bounded smooth domain and n 4

positive solutions for schrodinger system with asymptotically periodic potentials abstract this paper is concerned with the following class of elliptic equations u x u f u v 2 u in r n v x v g v u 2 v in r n where u v h 1 r n n 3 0 are coupling constants x and x are asymptotically periodic functions f and g are continuous functions with subcritical growth this type of system arises in particular in models in bose einstein condensates theory we prove the existence of positive solution for this weakly coupled system with 0 sufficiently large furthermore we obtain some sufficient conditions for the nonexistence of positive solutions

uniform holder bounds for nonlinear schrodinger systems with strong competition for the positive solutions of the gross pitaevskii system u u 1 u 3 u v 2 v v 2 v 3 u 2 v we prove that l boundedness implies c 0 boundedness for every 0 1 uniformly as moreover we prove that the limiting profile as is lipschitz continuous the proof relies upon the blowup technique and the monotonicity formulae by almgren and alt caffarelli and friedman this system arises in the hartree fock approximation theory for binary mixtures of bose einstein condensates in different hyperfine states extensions to systems with k 2 densities are given

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

interval oscillation of second order functional differential equations with oscillatory potentials abstract interval oscillation criteria are established for second order differential equations of the form e k t x t p t x g t q t x g t 1 x g t e t t 0 where 1 k p q e and g are continuous real valued functions k t 0 is nondecreasing g t is nondecreasing lim t g t we consider g t t and g t t separately illustrative examples are included

singularly perturbed elliptic systems and multi valued harmonic functions with free boundaries here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems the limiting problems involve multiple valued harmonic functions or in general harmonic maps to singular spaces and free interfaces between supports of various components of the maps the main results of the paper are the uniform lipschitz regularity of solutions as well as the regularity of free interfaces