multiple existence of sign changing solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of sign changing solutions of coupled nonlinear schrodinger equations p u 1 u 1 u 3 v 2 u in r 3 v 2 v 2 v 3 u 2 v in r 3 where 1 2 1 2 are positive and 0

on a class of singular elliptic systems we study the semilinear elliptic system u u p v q u 0in v u r v s v 0in u v 0on where rn n 1 is a smooth and bounded domain p q r s 0 under suitable ranges of exponents we obtain various results regarding the well posedness of our system

lane emden systems with negative exponents we study the elliptic system u u pv qin v u rv sin u v 0on in a bounded domain rn n 1 with a smooth boundary p s 0 and q r 0 we investigate the existence non existence and uniqueness of c2 c solutions in terms of p q r and s a necessary and sufficient condition for the c1 regularity of solutions up to the boundary is also obtained

optimal liouville type theorems for noncooperative elliptic schrodinger systems and applications we study multi component elliptic schrodinger systems arising in nonlinear optics and bose einstein condensation phenomena we prove new liouville type nonexistence theorems as well as a priori bounds decay and singularity estimates this is shown under an optimal sobolev growth restriction on the nonlinearities thus improving on recent results of dancer et al and of tavares et al these systems are of non cooperative form and hence cannot be tackled by maximum principle methods such as moving planes instead we rely on a delicate combination of rellich pohozaev type identities sobolev and interpolation inequalities on sn 1 and feedback and measure arguments we also extend our results to a rather general class of gradient type systems

infinitely many sign changing solutions for system of p laplace equations in rn abstract in this paper we consider the system of p laplacian equations in r n p u j j x u j p 2 u j i 1 k i j u i q u j q 2 u j in r n u j x 0 as x j 1 k where 1 p n max 2 p 2 q p n p n p i j are constants i j 1 k j j 0 j 1 2 k i j 0 i j i j 1 k j j 1 k are the potential functions and satisfy suitable decay assumptions the existence of infinitely many sign changing solutions is proved by the truncation method and by the concentration compactness analysis on the approximating solutions

self similarity and asymptotic stability for coupled nonlinear schrodinger equations in high dimensions abstract this paper is concerned with systems of coupled schrodinger equations with polynomial nonlinearities and dimension n 1 we show the existence of global self similar solutions and prove that they are asymptotically stable in a framework based on weak l p spaces whose elements have local finite l 2 mass the radial symmetry of the solutions is also addressed

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

multiple existence of solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of solutions of the coupled nonlinear schrodinger equations p u 1 u u 3 v 2 u in r 3 v 2 v v 3 u 2 v in r 3 where 1 2 and 0

bifurcation diagrams of coupled schrodinger equations radially symmetric solutions of many important systems of partial differential equations can be reduced to systems of special ordinary differential equations a numerical solver for initial value problems for such systems is developed based on matlab and numerical bifurcation diagrams are obtained according to the behavior of the solutions various bifurcation diagrams of coupled schrodinger equations from nonlinear physics are obtained which suggests the uniqueness of the ground state solution

optimal entanglement witnesses from generalized reduction and robertson maps we provide a generalization of the reduction and robertson positive maps in matrix algebras they give rise to a new class of optimal entanglement witnesses their structural physical approximation is analyzed as a byproduct we provide new examples of ppt positive partial transpose entangled states

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

existence of multiple solutions to a class of nonlinear schrodinger system with external sources terms abstract we study a class of nonlinear schrodinger system with external sources terms as perturbations in order to obtain existence of multiple solutions this system arises from bose einstein condensates etc as these external sources terms are positive functions and small in some sense we use nehari manifold to get the existence of a positive ground state solution and a positive bound state solution

solutions with a prescribed number of zeros for nonlinear schrodinger systems abstract we find an interval of some parameters where there exists a sign changing solution of nonlinear schrodinger systems having a prescribed number of zeros

multi bump bound states for a system of nonlinear schrodinger equations abstract motivated by problems arising in nonlinear optics and bose einstein condensates we consider in r n with n 3 the following system of coupled schrodinger equations u i v i x u i u i j 1 d i j u j 2 u i 0 lim x u i x 0 i 1 d where 0 is a parameter i j j i are positive constants and v i non negative given potentials we assume that the interior of i 1 d v i 1 0 admits m connected components 1 m which are of class c 1 and isolated in each v i 1 0 for each non empty j 1 m we prove that the system admits for any large a multi bump solution u r n r d which is small in r n j j j and on each j j j close in h 1 norm to a least energy solution of the limit problem u i u i j 1 d i j u j 2 i 1 d subjected to homogeneous dirichlet boundary condition an explicit condition on the matrix i j is given to ensure our solutions have at least two positive components

a minimal energy tracking method for non radially symmetric solutions of coupled nonlinear schrodinger equations we aim at developing methods to track minimal energy solutions of time independent m component coupled discrete nonlinear schrodinger dnls equations we first propose a method to find energy minimizers of the 1 component dnls equation and use it as the initial point of the m component dnls equations in a continuation scheme we then show that the change of local optimality occurs only at the bifurcation points the fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve by combining all these techniques with a parameter switching scheme we successfully compute a non radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

steady state solutions for a general activator inhibitor model abstract we discuss existence results for a singular gierer meinhardt elliptic system with zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology and has interesting and challenging mathematical properties the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the general activator inhibitor model

a sharp threshold of blow up for coupled nonlinear schrodinger equations this paper is concerned with the coupled supercritical nonlinear schrodinger equations which have applications in many physical problems especially in nonlinear optics two types of new invariant evolution flows are established a sharp threshold of blow up and global existence of solutions for the equations is derived it is shown that the main result obtained includes parts of those presented by fanelli and montefusco 2007 j phys a math theor 40 14139 50x

existence and asymptotic behavior of vector solutions for coupled nonlinear kirchhoff type systems abstract this paper deals with the following linearly coupled nonlinear kirchhoff type system a 1 b 1 r 3 u 2 d x u 1 u f u v in r 3 a 2 b 2 r 3 v 2 d x v 2 v g v u in r 3 u v h 1 r 3 where a i 0 b i 0 i 0 are constants for i 1 2 0 is a parameter and f g c r r under the general berestycki lions type assumptions on f and g we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods we also study the asymptotic behavior of these solutions as 0

ground states of two component attractive bose einstein condensates i existence and uniqueness abstract we study ground states of two component bose einstein condensates bec with trapping potentials in r 2 where the intraspecies interaction a 1 a 2 and the interspecies interaction are both attractive i e a 1 a 2 and are all positive the existence and non existence of ground states are classified completely by investigating equivalently the associated l 2 critical constraint variational problem the uniqueness and symmetry breaking of ground states are also analyzed under different types of trapping potentials as a a a 1 a a 2 where 0 a i a w 2 2 i 1 2 is fixed and w is the unique positive solution of w w w 3 0 in r 2 the semi trivial limit behavior of ground states is tackled in the companion paper 12

uniformly bounded weak and classical solutions to a singular parabolic system and applications abstract equations of both elliptic and parabolic type featuring singular nonlinearities have appeared in numerous works throughout the years in this work we consider a time dependent problem featuring nonlinearities of the form u p v q u r v s subject to homogeneous dirichlet boundary conditions and prove the existence of uniformly bounded weak and classical solutions under appropriate conditions on p q r s these results can in some ways be seen as a generalization of the results presented in 9 these results are obtained using a functional method motivated by works found in 8 22 6 etc and the boundary behaviour of a fundamental singular elliptic equation described in 16

uniqueness of minimal blow up solutions to nonlinear schrodinger system abstract in this paper we are concerned with the finite time blow up solutions to the following n coupled nonlinear schrodinger system in r 2 0 1 i t j j k 1 n a j k k 2 j 0 where n 2 and the coefficient matrix a satisfies a j k a k j 0 we study the properties of the blow up solutions which obtain the minimal l 2 norm and prove that such solution is unique up to the symmetries of the system

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

positive solutions of lane emden systems with negative exponents existence boundary behavior and uniqueness abstract we study the existence boundary behavior and uniqueness of solutions for the singular elliptic system u u p v q v u r v s u 0 v 0 x u v 0 where is a bounded domain with smooth boundary in r n p s 0 and q r 0 our results are obtained in a range of p q r s different from those in m ghergu lane emden systems with negative exponents j funct anal 258 2010 3295 3318

existence uniqueness and asymptotic behavior of solutions for a singular parabolic equation abstract in this paper we are concerned with a singular parabolic equation v t v f x t v 2 v in a smooth bounded domain r n subject to zero dirichlet boundary condition and initial condition 0 under the assumptions on and f x t some existence and uniqueness results are obtained by applying parabolic regularization method and the sub supersolutions method we also discuss the asymptotic behaviors of solutions in the sense of l 2 0 t w 0 1 2 and l 0 t l 2 norms as 0 or as a byproduct we obtain the existence of solutions for some problems which blow up on the boundary

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

a series of absolutely indecomposable positive maps in matrix algebras abstract we provide examples of positive maps in m n c n 4 which cannot be decomposed into a sum of a 2 positive map and a 2 copositive map

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

on the classification of standing wave solutions to a coupled schrodinger system in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball and entire space as the parameters within certain regions in addition a complete structure of different types of solutions for the radial case is also provided

the structural physical approximation conjecture it was conjectured that the structural physical approximation spa of an optimal entanglement witness is separable or equivalently that the spa of an optimal positive map is entanglement breaking this conjecture was disproved first for indecomposable maps and more recently for decomposable maps the arguments in both cases are sketched along with important related results this review includes background material on topics including entanglement witnesses optimality duality of cones decomposability and the statement and motivation for the spa conjecture so that it should be accessible for a broad audience

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

two coupled nonlinear schrodinger equations involving a non constant coupling coefficient abstract in this paper we study the existence and multiplicity of vector positive solutions for two coupled nonlinear schrodinger equations involving a non constant coupling coefficient

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

stationary solitons of a three wave model generated by type ii second harmonic generation in quadratic media abstract in this paper we prove the uniqueness of stationary standing wave solutions of an optical model generated by type ii second harmonic generation shg with behaviors tending to zero at infinity under certain conditions on parameters in addition we provide the same issues for the dirichlet boundary value problems on the ball centered at the origin a classification of solutions for radial case is also established

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

a liouville s theorem for a fractional elliptic system with indefinite nonlinearities abstract in this paper we consider a fractional elliptic system with indefinite nonlinearities we prove that every positive bounded solution of the system is monotone increasing along x 1 direction by using the direct method of moving planes based on this monotonicity we obtain the nonexistence of positive solutions and a liouville type results for the fractional elliptic system as an application we get the liouville type results for the fractional elliptic system with cubic nonlinearities

a natural constraint approach to normalized solutions of nonlinear schrodinger equations and systems abstract the paper deals with the existence of normalized solutions to the system u 1 u 1 u 3 u v 2 in r 3 v 2 v 2 v 3 u 2 v in r 3 r 3 u 2 a 1 2 and r 3 v 2 a 2 2 for any 1 2 a 1 a 2 0 and 0 prescribed we present a new approach that is based on the introduction of a natural constraint associated to the problem we also show that as phase separation occurs for the solutions that we find our method can be adapted to scalar nonlinear schrodinger equations with normalization constraint and leads to alternative and simplified proofs to some results already available in the literature

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

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

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

stability and instability of standing waves to a system of schrodinger equations with combined power type nonlinearities abstract we obtain the existence of standing wave solutions to a coupled nonlinear schrodinger system i u 1 t u 1 u 1 p k 1 m k u 1 p k 2 u 2 q k u 1 in r n 0 t i u 2 t u 2 u 2 p k 1 m k u 1 p k u 2 q k 2 u 2 in r n 0 t u 1 x 0 u 10 x u 2 x 0 u 20 x in r n where 0 p 4 n 2 and k k p k q k are real constants with 1 p k p 1 1 q k p 1 p k q k p 2 and k q k k p k k 1 m we also discuss the stability and instability of the standing wave solutions

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

spike layer solutions to singularly perturbed semilinear systems of coupled schrodinger equations abstract let be a bounded domain in r n n 3 we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple schrodinger equations in se e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 where 1 2 are positive constants we prove that under neumann or dirichlet boundary conditions for any e 0 when min 1 2 or max 1 2 system se has a least energy solution u e v e and when min 1 2 max 1 2 system se has no solution suppose p e q e are the local maximum points of u e v e respectively then under neumann boundary conditions as e small enough both of p e q e locate on the boundary of furthermore when 0 p e q e e 0 as e 0 and for n 2 and n 3 p e q e converge to the same point on the boundary which is the maximum point of mean curvature of the boundary however when 0 p e q e e as e 0 and suppose p e p and q e q then for n 2 and n 3 p q must be the maximum points of the mean curvature on the boundary and p q might be a same point if the mean curvature of the boundary has only one maximum point under dirichlet boundary conditions we can prove that as long as the least energy solution se exists the same asymptotic behavior of the least energy solution u e v e holds as described in lin and wei 2005 10 for 0 or for 0 thus our results are an extension of the results in lin and wei 2005 10

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

positive solutions to some systems of coupled nonlinear schrodinger equations abstract we study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time independent schrodinger equations u j j u j j u j 3 k 1 k j n j k u j u k 2 u j w 1 2 r n j 1 n where n 1 2 3 j j 0 for j 1 n the coupling parameters j k k j r for j k 1 n j k precisely we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors additionally with more restrictive conditions on the coupled parameters we show that the bound states founded are positive

sharp thresholds of blow up and global existence for the coupled nonlinear schrodinger system in this paper we establish two new types of invariant sets for the coupled nonlinear schrodinger system in the euclidean n space rn and derive two sharp thresholds of blow up and global existence for its solutions some analogous results for the nonlinear schrodinger system posed on the hyperbolic space hn and on the standard 2 sphere s2 are also presented our arguments and constructions are improvements of some previous works on this direction at the end we give some heuristic analysis about the strong instability of the solitary waves the relation between the two types of thresholds is a very interesting problem and we leave it as an open problem for further study

global existence for a singular gierer meinhardt system abstract this paper is concerned with existence results for a singular gierer meinhardt system subject to zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology the mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure we use a functional method to obtain both upper and lower bounds for the perturbed system and then use sobolev embedding theorem to prove the existence of a pair of positive solutions under suitable conditions this method is first used in a singular parabolic system and is completely different than the traditional methods of sub and super solutions

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

on structural physical approximations and entanglement breaking maps very recently a conjecture saying that the so called structural physical approximations spas to optimal positive maps optimal entanglement witnesses give entanglement breaking eb maps separable states has been posed korbicz et al 2008 phys rev a 78 062105 the main purpose of this contribution is to explore this subject first we extend the set of entanglement witnesses supporting the conjecture then we ask whether spas constructed from other than the depolarizing channel maps also lead to eb maps and show that in general this is not the case on the other hand we prove an interesting fact that for any positive map there exists an eb channel such that the spa of constructed with the aid of is again an eb channel finally we ask similar questions in the case of continuous variable systems we provide a simple way of constructing spa and prove that in the case of the transposition map it gives the eb channel

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 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

two component bose einstein condensates abstract recently coupled systems of nonlinear schrodinger equations have been used extensively to describe bose einstein condensates in this paper we study the structure of vortices of the coupled nonlinear equations for two component bose einstein condensates bec in a three dimensional space we show that vortices is 1 rectifiable set and give its mean curvature in particular we show that large interspecies scattering length causes vortices for two component bec

blowup and solitary wave solutions with ring profiles of two component nonlinear schrodinger systems abstract blowup ring profiles have been investigated by finding non vortex blowup solutions of nonlinear schrodinger equations nlses cf fibich et al 2005 7 and fibich et al 2007 8 however those solutions have infinite l 2 norm so one may not maintain the ring profile all the way up to the singularity to find h 1 non vortex blowup solutions with ring profiles we study the blowup solutions of two component systems of nlses with nonlinear coefficients and j j 1 2 when 0 and 1 2 0 the two component system can be transformed into a multi scale system with fast and slow variables which may produce h 1 blowup solutions with non vortex ring profiles we use the localized energy method with symmetry reduction to construct these solutions rigorously on the other hand these solutions may describe steady non vortex bright ring solitons various types of ring profiles including m ring and ring ring profiles are presented by numerical solutions

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

solitary and self similar solutions of two component system of nonlinear schrodinger equations conventionally to learn wave collapse and optical turbulence one must study finite time blow up solutions of one component self focusing nonlinear schrodinger equations nlse here we consider simultaneous blow up solutions of two component systems of self focusing nlse by studying the associated self similar solutions we prove two components of solutions blow up at the same time these self similar solutions may come from solitary wave solutions with multi bumps forming abundant geometric patterns which cannot be found in one component self focusing nlse our results may provide the first step to investigate optical turbulence in two component systems of nlse

semitrivial vs fully nontrivial ground states in cooperative cubic schrodinger systems with d 3 equations abstract in this work we consider the weakly coupled schrodinger cubic system u i i u i i u i 3 u i j i b i j u j 2 u i h 1 r n r i 1 d where 1 n 3 i i 0 and b i j b j i 0 for i j this system admits semitrivial solutions that is solutions u u 1 u d with null components we provide optimal qualitative conditions on the parameters i i and b i j under which the ground state solutions have all components nontrivial or conversely are semitrivial this question had been clarified only in the d 2 equations case for d 3 equations prior to the present paper only very restrictive results were known namely when the above system was a small perturbation of the super symmetrical case i and b i j b we treat the general case uncovering in particular a much more complex and richer structure with respect to the d 2 case

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

bounded solutions to a singular parabolic system abstract in this paper we are concerned with the singular parabolic system u t u f x v p v t v g x u q in a smooth bounded domain r n subject to zero dirichlet conditions with initial conditions u 0 x v 0 x 0 this problem is of interest as it is related to some problems in biology and physics under suitable assumptions on p q and f x g x some existence results of weak and classical solutions are obtained using a functional method this method is motivated by such results found in 4 and 5 when dealing with singular parabolic systems and the related references within

segregated and synchronized vector solutions for nonlinear schrodinger systems we consider the following nonlinear schrodinger system in mathbb r 3 left begin array ll delta u p x u mu u 2 u beta v 2u quad x in mathbb r 3 delta v q x v nu v 2 v beta u 2v quad x in mathbb r 3 end array right where p r and q r are positive radial potentials mu 0 nu 0 and beta in mathbb r is a coupling constant this type of system arises in particular in models in bose einstein condensates theory

symmetric ground state solutions of m coupled nonlinear schrodinger equations abstract we prove the existence of radial and radially decreasing ground states of an m coupled nonlinear schrodinger equation with a general nonlinearity

on strongly indefinite systems involving the fractional laplacian abstract this paper is devoted to study strongly indefinite systems involving the fractional laplacian on bounded domains explicitly we obtain existence and non existence results a p r i o r i estimates of gidas spruck type and a symmetry result in addition we give a different proof for the a p r i o r i estimate for nonlinear elliptic problems with the fractional laplacian obtained in cabre and tan 2010 and tan 2013

indecomposable positive maps in low dimensional matrix algebras abstract we classify a series of positive maps in low dimensional matrix algebras with respect to the degree of indecomposability and provide stronger indecomposable maps in m n we also present a basic result about the atomic property

multi peak solutions to coupled schrodinger systems with neumann boundary conditions abstract in this paper we are concerned with the following two coupled schrodinger systems in a bounded domain r n n 2 3 with neumann boundary conditions e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 u n 0 v n 0 on suppose the mean curvature h p of the boundary has several local minimums or local maximums we obtain the existence of solutions with multi peaks to the system with all peaks being on the boundary and all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain

blow up behavior of ground states for a nonlinear schrodinger system with attractive and repulsive interactions abstract we consider a nonlinear schrodinger system arising in a two component bose einstein condensate bec with attractive intraspecies interactions and repulsive interspecies interactions in r 2 we get ground states of this system by solving a constrained minimization problem for some kinds of trapping potentials we prove that the minimization problem has a minimizer if and only if the attractive interaction strength a i i 1 2 of each component of the bec system is strictly less than a threshold a furthermore as a 1 a 2 a a the asymptotical behavior for the minimizers of the minimization problem is discussed our results show that each component of the bec system concentrates at a global minimum of the associated trapping potential

multiplicity of localized solutions of nonlinear schrodinger systems for infinite attractive case abstract in this paper we consider the following nonlinear schrodinger system 2 u v 1 x u u v 2 in r n u h 1 r n 2 v v 2 x v u 2 v in r n v h 1 r n where n 3 0 1 v i c 1 r n r 0 inf x r n v i x sup x r n v i x i 1 2 and there exists a bounded open set r n with smooth boundary such that inf x min v 1 x v 2 x 0 we prove the existence of a given number of semi positive solutions localized in here we say u v is semi positive if either u 0 or v 0

note on ground states of a nonlinear schrodinger system abstract we give a sufficient condition for the existence of positive radial ground states of the time independent schrodinger system u 1 u 1 1 u 1 3 12 u 2 2 u 1 13 u 3 2 u 1 in r n u 2 u 2 12 u 1 2 u 2 2 u 2 3 23 u 3 2 u 2 in r n u 3 u 3 13 u 1 2 u 3 23 u 2 2 u 3 3 u 3 3 in r n u 1 x 0 u 2 x 0 u 3 x 0 as x where n 1 2 3 0 j 0 and i j 0 i j for i j 1 2 3 and in some special cases our conditions are also necessary

the structural physical approximations and optimal entanglement witnesses we introduce the notions of positive and copositive types for entanglement witnesses depending on the distance to the positive part and copositive part an entanglement witness w is of positive type if and only if its partial transpose w is of copositive type we show that if the structural physical approximation spa of w is separable then w should be of copositive type and the spa of w is never separable unless w is of both positive and copositive types this shows that the spa conjecture is meaningful only for those of copositive type we provide examples to show that the spa conjecture fails even for the case of copositive types

infinitely many solutions for the perturbed bose einstein condensates system abstract we consider the perturbed nonlinear elliptic system as following u j j u j i 1 2 i j u i 2 u j j x u j h 0 1 j 1 2 where is a bounded smooth domain in r n n 3 and j 1 1 is the first eigenvalue of with the dirichlet boundary condition the perturbation terms j l 2 for j 1 2 assume that the matrix i j i j 1 2 is either positive definite when n 1 2 or anti symmetric when n 3 we show the existence of infinitely many solutions to this system the results are also true for the m coupled system i e i j 1 m m 2