existence and uniqueness of multi bump bound states of nonlinear schrodinger equations with electromagnetic fields abstract in this paper we are concerned with the existence and uniqueness of multi bump bound states of the nonlinear schrodinger equations with electromagnetic potential i t i a x 2 v x p 2 x r n for sufficiently small 0 where i is the imaginary unit 2 p 2 n n 2 for n 3 and 2 p for n 1 2 v x is a bounded real function on r n and a x a 1 x a 2 x a n x is such that a j x is a bounded real function on r n for j 1 2 n for any finite collection of a 1 a 2 a k of non degenerate critical points of v x we show that there exists a solution x t exp iet u x which is a small perturbation of a sum of one bump solutions concentrated at a 1 a 2 a k and u x is unique up to rotations for small 0

dirichlet and neumann problems for klein gordon maxwell systems abstract this paper deals with the klein gordon maxwell system in a bounded spatial domain we study the existence of solutions having a specific form namely standing waves in equilibrium with a purely electrostatic field we prescribe dirichlet boundary conditions on the matter field and either dirichlet or neumann boundary conditions on the electric potential

ground state solutions for the electrostatic nonlinear klein gordon maxwell system abstract in this paper we study the nonlinear klein gordon equation coupled with the maxwell equation in the electrostatic case p u m 2 e 2 u f u in r 3 e e u 2 in r 3 where m e 0 benci and fortunato 2002 3 and d aprile and mugnai 2004 6 showed that for any u h 1 r 3 the second equation of problem p has a unique solution u d 1 2 r 3 the map u h 1 r 3 u d 1 2 r 3 is continuously differentiable and u e 0 furthermore we prove that max e u u u 0 where u u u 2 then we consider the ground state solution of problem p with f u u p 2 u 2 p 6

applications of dualizing complexes to buchsbaum rings let a m k denote a local noetherian ring with d dim a admitting a normalized dualizing complex d by well known theorems on local duality a is cohen macaulay ring if and only if the d th place truncated dualizing complex r d is isomorphic to the zero complex in the derived category d a see section 1 for more precise statements in the following we consider a class of local rings for which the truncated dualizing complex t d is isomorphic to a complex of vector spaces over a m k there have been several efforts to get results about local rings which generalize the cohen macaulay property see for instance 7 13 14 a connection to our consideration is given in the following sense in 2 3 we prove one of our main theorems the complex t d of a is isomorphic to a complex of vector spaces over k if and only if a is buchsbaum ring in the sense of 13 141 they are those rings for which the difference l a d e x a between length and multiplicity of a system of parameters x does ndt depend on x see also 2 1 for a brief summary of essential facts about buchsbaum rings and relation to a conjecture of d a buchsbaum in 7 m hochster and j l roberts have studied purity of the frobenius map and related things to show that the local cohomology modules of certain rings of such type behave nicely applying our criterion 3 1 to this situation we can show in 4 1 that such rings have a truncated dualizing complex t d which is isomorphic to a complex of vector spaces over k following the examples given in 7 we get a number of buchsbaum rings for instance let vc ip be a projective variety defined by an ideal which is generated by square free monomials then v is arithmetically buchsbaum if and only if v is a cohen macaulay variety our theorem 2 3 is related to a result of r kiehl in 8 using the surjective map criterion of 141 he showed that the canonical module of a is buchsbaum if t d is isomorphic to a complex of vector spaces over k this yields in 8 that the local rings at the cusps of hilbert modular groups are factorial buchsbaum rings which are not cohen macaulay 61 oool 8708 82 040061 17 05 00 o

on the determination of the basin of attraction of a periodic orbit in two dimensional systems we consider the general nonlinear differential equation x f x with x epsilon r 2 and develop a method to determine the basin of attraction of a periodic orbit borg s criterion provides a method to prove existence uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction in order to use the criterion one has to find a function w epsilon c 1 r 2 r such that l w x w x l x is negative for all x epsilon k where k is a positively invariant set here l x is a given function and w x denotes the orbital derivative of w in this paper we prove the existence and smoothness of a function w such that l w x mu parallel to f x parallel to we approximate the function w which satisfies the linear partial differential equation w x mu parallel to f x parallel to l x using radial basis functions and obtain an approximation omega such that l omega x 0 using radial basis functions again we determine a positively invariant set k so that we can apply borg s criterion as an example we apply the method to the van der pol equation

layered solutions for a semilinear elliptic system in a ball abstract we consider the following system of schrodinger poisson equations in the unit ball b 1 of r 3 e 2 v v v v p 0 4 v 2 with the dirichlet boundary conditions v 0 on b 1 here e 0 v b 1 r 1 p 7 3 we exhibit a family of positive radially symmetric solutions v e e such that v e concentrates on a sphere in the interior of b 1 as e 0 our approach relies upon a finite dimensional reduction by the lyapunov schmidt method which allows us to reduce the problem to maximizing a one dimensional functional the solution of such reduced problem also provides the radius of the concentration sphere

infinitely many solutions and least energy solutions for klein gordon maxwell systems with general superlinear nonlinearity abstract this paper is concerned with the following klein gordon maxwell system u v x u 2 u f x u x r 3 u 2 x r 3 where 0 is a constant v c r 3 r f c r 3 r r and f is superlinear at infinity using some weaker superlinear conditions instead of the common super cubic conditions on f we prove that the above system has 1 infinitely many solutions when v x is coercive and sign changing 2 a least energy solution when v x is positive periodic these results improve the related ones in the literature

schrodinger poisson equations without ambrosetti rabinowitz condition abstract we prove the existence of ground state solutions for a stationary schrodinger poisson equation in r 3 the proof is based on the mountain pass theorem and it does not require the ambrosetti rabinowitz condition

multiple solutions for superlinear schrodinger poisson system with sign changing potential and nonlinearity in this paper we study the following schrodinger poisson system u v x u x u f x u in r 3 u 2 in r 3 where the nonlinearity f and the potential v are allowed to be sign changing under some appropriate assumptions on v and f we obtain the existence of two different solutions of the system via the ekeland s variational principle and the mountain pass theorem

on the determination of the basin of attraction of periodic orbits in three and higher dimensional systems the determination of the basin of attraction of a periodic orbit can be achieved using a lyapunov function a lyapunov function can be constructed by approximation of a first order linear pde for the orbital derivative via meshless collocation however if the periodic orbit is only accessible numerically a different method has to be used near the periodic orbit borg s criterion provides a method to obtain information about the basin of attraction by measuring whether adjacent solutions approach each other with respect to a riemannian metric using a numerical approximation of the periodic orbit and its first variation equation a suitable riemannian metric is constructed

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

positive ground state solutions for the critical klein gordon maxwell system with potentials abstract this paper deals with the klein gordon maxwell system when the nonlinearity exhibits critical growth we prove the existence of positive ground state solutions for this system when a periodic potential v is introduced the method combines the minimization of the corresponding euler lagrange functional on the nehari manifold with the brezis and nirenberg technique

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

canonical elements in local cohomology modules and the direct summand conjecture one of the objectives of this paper is to show that the usual homological consequences of the existence of big cohen macaulay henceforth c m modules e g the new intersection conjecture of peskine szpiro and roberts and the evans griffith syzygy conjecture follow from the direct summand conjecture when the residual characteristic of the local ring is positive this gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive and reduces the general case of all these conjectures to one rather down to earth conjecture of course this places the direct summand conjecture in a position of central importance so that it now merits an allout attack some partial results on this problem are given in section 6 the conjecture asserts that a regular noetherian ring r is a direct summand as an r module of every module finite extension ring s 3 r the other main objective of this paper is to formulate and develop a theory of certain canonical elements in the local cohomology of special modules of syzygies neither the modules of syzygies nor the induced maps between them are canonical but the identification between the canonical elements is independent of the choices in particular a canonical element qr e hz syz k is associated see section 3 for details with each ndimensional local ring r m k rings are commutative associative with identity local ring means noetherian ring with a unique maximal ideal the conjecture that qr is nonzero for all local rings r turns out to be equivalent to the direct summand conjecture for a given r an infinite family of cases of the latter implies the former

low energy solutions for singularly perturbed coupled nonlinear systems on a riemannian manifold with boundary abstract let m g be a smooth compact riemannian manifold with smooth boundary with n dim m 2 3 we suppose the boundary m to be a smooth submanifold of m with dimension n 1 we consider a singularly perturbed nonlinear system namely klein gordon maxwell proca system or klein gordon maxwell system of schroedinger maxwell system on m we prove that the number of low energy solutions when the perturbation parameter is small depends on the topological properties of the boundary m by means of the lusternik schnirelmann category also these solutions have a unique maximum point that lies on the boundary

on the schrodinger poisson slater system behavior of minimizers radial and nonradial cases this paper is motivated by the study of a version of the so called schrodinger poisson slater problem delta u omega u lambda left u 2 star frac 1 x right u u p 2 u where u in h 1 mathbb r 3 we are concerned mostly with p in 2 3 the behavior of radial minimizers motivates the study of the static case 0 among other things we obtain a general lower bound for the coulomb energy which could be useful in other frameworks the radial and nonradial cases turn out to yield essentially different situations

on the planar schrodinger poisson system with the axially symmetric potential abstract in this paper we develop some new variational and analytic techniques to prove that the following planar schrodinger poisson system u v x u u f u x r 2 u 2 x r 2 admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space where v x is axially symmetric our results improve and extend the ones in the case v 1 and f u u p 2 u with 2 p 6 in particular we use the assumption that 2 v x v x x is bounded from below instead of the usually one that lim x v x 1 moreover v x is even admitted to be unbounded

on the direct summand conjecture and its derived variant andre recently gave a beautiful proof of hochster s direct summand conjecture in commutative algebra using perfectoid spaces his two main results are a generalization of the almost purity theorem the perfectoid abhyankar lemma and a construction of certain faithfully flat extensions of perfectoid algebras where discriminants acquire all p power roots in this paper we explain a quicker proof of hochster s conjecture that circumvents the perfectoid abhyankar lemma instead we prove and use a quantitative form of scholze s hebbarkeitssatz the riemann extension theorem for perfectoid spaces the same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de jong

ground state solutions for generalized quasilinear schrodinger equations with variable potentials and berestycki lions nonlinearities by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature

solutions with multiple peaks for nonlinear elliptic equations solutions with peaks near the critical points of q x are constructed for the problem delta u lambda 2 u q x u p 2 u x is an element of r n u is an element of h 1 and 2 p 2n n 2

converse theorems on contraction metrics for an equilibrium the stability and basin of attraction of an equilibrium can be determined by a contraction metric a contraction metric is a riemannian metric with respect to which the distance between adjacent trajectories decreases the advantage of a contraction metric over e g a lyapunov function is that the contraction condition is robust under perturbations of the system while the sufficiency of a contraction metric for the existence stability and basin of attraction of an equilibrium has been extensively studied in this paper we will prove converse theorems showing the existence of several different contraction metrics this will be useful to develop algorithms for the construction of contraction metrics

an intrinsic observer for a class of lagrangian systems we propose a new design method of asymptotic observers for a class of nonlinear mechanical systems lagrangian systems with configuration position measurements our main contribution is to introduce a state position and velocity observer that is invariant under any changes of the configuration coordinates the observer dynamics equations as the euler lagrange equations are intrinsic the design method uses the riemannian structure defined by the kinetic energy on the configuration manifold the local convergence is proved by showing that the jacobian of the observer dynamics is negative definite contraction for a particular metric defined on the state space a metric derived from the kinetic energy and the observer gains from a practical point of view such intrinsic observers can be approximated when the estimated configuration is close to the true one by an explicit set of differential equations involving the riemannian curvature tensor these equations can be automatically generated via symbolic differentiations of the metric and potential up to order two numerical simulations for the ball and beam system an example where the scalar curvature is always negative show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics

multiple positive solutions for a schrodinger poisson slater system abstract in this paper we investigate the existence of positive solutions to the following schrodinger poisson slater system u u u u p 2 u in u 2 in u 0 on where is a bounded domain in r 3 is a fixed positive parameter and p 2 2 n n 2 we prove that if p is near the critical sobolev exponent 2 then the number of positive solutions is greater then the lusternik schnirelmann category of

brief paper stability and robustness analysis of nonlinear systems via contraction metrics and sos programming a wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities lmis these techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares sos programming in this paper we further extend the class of systems that can be analyzed with convexity based methods we show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics via contraction analysis and sos programming since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system we develop an algorithm for finding such contraction metrics using sos programming the search process is made computationally tractable by relaxing matrix definiteness constraints the feasibility of which indicates the existence of a contraction metric to sos constraints on polynomial matrices we illustrate our results through examples from the literature and show how our contraction based approach offers advantages when compared with traditional lyapunov analysis

uniqueness of positive multi lump bound states of nonlinear schrodinger equations in this paper we are concerned with multi lump bound states of the nonlinear schrodinger equation

necessary conditions for a limit cycle and its basin of attraction in this paper we consider a general differential equation of the form x f x with f epsilon c l r n r n and n greater than or equal to 2 borg hartman leonov and others have studied sufficient conditions for the existence uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction they used a certain contraction property of the flow with respect to the euclidian or a riemannian metric in this paper we also prove sufficient conditions including upper bounds for the floquet exponents of the periodic orbit moreover we show the necessity of these conditions using floquet theory and a lyapunov function

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

ground state sign changing solutions for kirchhoff type problems in bounded domains abstract in the present paper we consider the existence of ground state sign changing solutions for a class of kirchhoff type problems 0 1 a b u 2 d x u f u x u 0 x where r n is a bounded domain with a smooth boundary n 1 2 3 a 0 b 0 and f c r r under some weak assumptions on f with the aid of some new analytical skills and non nehari manifold method we prove that 0 1 possesses one ground state sign changing solution u b and its energy is strictly larger than twice that of the ground state solutions of nehari type furthermore we establish the convergence property of u b as the parameter b 0 our results improve and generalize some results obtained by w shuai 2015 34

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

construction of a cpa contraction metric for periodic orbits using semidefinite optimization abstract a riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction while the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit the explicit construction of the metric is a difficult problem in this paper the construction of such a contraction metric is achieved by formulating it as an equivalent problem namely a feasibility problem in semidefinite optimization the contraction metric a matrix valued function is constructed as a continuous piecewise affine cpa function which is affine on each simplex of a triangulation of the phase space the contraction conditions are formulated as conditions on the values at the vertices the paper states a semidefinite optimization problem we prove on the one hand that a feasible solution of the optimization problem determines a cpa contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough an objective function can be used to obtain a bound on the largest floquet exponent of the periodic orbit

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

a minimization method and applications to the study of solitons abstract roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time a soliton is a solitary wave which exhibits some strong form of stability so that it has a particle like behavior in this paper we prove a general abstract theorem theorem 26 which allows to prove the existence of a class of solitons such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons then we apply the abstract theory to problems related to the nonlinear schrodinger equation nse and to the nonlinear klein gordon equation nkg

convergence properties for discrete time nonlinear systems three similar convergence notions are considered two of them are the long established notions of convergent dynamics and incremental stability the other is the more recent notion of contraction analysis all three convergence notions require that all solutions of a system converge to each other in this note we investigate the differences between these convergence properties for discrete time and time varying nonlinear systems by comparing the properties in pairs and using examples we also demonstrate a time varying smooth lyapunov function characterization for each of these convergence notions and with appropriate assumptions we provide several sufficient conditions to establish relationships between these properties in terms of lyapunov functions

solitary waves for the klein gordon maxwell system with critical exponent abstract in this paper we study the existence of solutions for nonlinear klein gordon maxwell equations coupled with maxwell s equations when the nonlinearity exhibits critical growth we improve some previous existence results in azzollini et al 2009 5 carriao et al 2009 4 and cassani 2004 3

multiple solutions for the nonhomogeneous klein gordon equation coupled with born infeld theory on r3 abstract in this paper we prove the existence of multiple solutions for the following nonhomogeneous klein gordon equation coupled with born infeld theory u m 2 2 u u q 2 u h x in r 3 4 4 u 2 in r 3 when 2 q 6

the existence and concentration of positive solutions for a nonlinear schrodinger poisson system with critical growth we consider the schrodinger poisson system e2 u v x u x u f u u2 in r3 where the nonlinear term f is of critical growth in this paper we construct a solution u of the above elliptic system which concentrates at an isolated component of positive locally minimum points of v as 0 under certain conditions on f in particular the monotonicity of f s s3 and the so called ambrosetti rabinowitz condition are not required

existence of multiple solutions of kirchhoff type equation with sign changing potential abstract in this paper we study the existence of infinitely many nontrivial solutions for a class of nonlinear schrodinger kirchhoff type equation a b r n u 2 dx u v x u f x u in r n u x 0 as x where constants a 0 b 0 and the potential v x is allowed to be sign changing under general superlinear assumption on nonlinearity f x u we establish the existence of infinitely many solutions via variational methods which unifies and improves the recent results of wu 2011 9

three dimensional vortices in abelian gauge theories abstract in this paper we consider an abelian gauge theory in r 4 equipped with the minkowski metric this theory leads to a system of equations the klein gordon maxwell equations which provide models for the interaction between the electromagnetic field and matter a three dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid under suitable assumptions we prove the existence of vortex solutions

number and profile of low energy solutions for singularly perturbed klein gordon maxwell systems on a riemannian manifold abstract given a 3 dimensional riemannian manifold m g we investigate the existence of positive solutions of the klein gordon maxwell system e 2 g u a u u p 1 2 q v 1 2 u in m g v 1 q 2 u 2 v q u 2 in m and schrodinger maxwell system e 2 g u u u v u p 1 in m g v v q u 2 in m when p 4 6 we prove that the number of one peak solutions depends on the topological properties of the manifold m by means of the lusternik schnirelmann category

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

coupled klein gordon and born infeld type equations looking for solitary waves the existence of infinitely many non trivial radially symmetric solitary waves for the nonlinear klein gordon equation coupled with a born infeld type equation is established under general assumptions

existence of solitary wave solutions for the nonlinear klein gordon equation coupled with born infeld theory with critical sobolev exponent abstract in this paper we prove the existence of solutions for the nonlinear klein gordon equation coupled with born infeld theory under the electrostatic solitary wave ansatz by using variational methods

on a system involving a critically growing nonlinearity abstract this paper deals with the system u u q u 3 u in b r q u 5 in b r u 0 on b r we prove existence and nonexistence results depending on the value of

on critical klein gordon maxwell systems with super linear nonlinearities abstract this paper is concerned with two classes of critical klein gordon maxwell systems as follows u v x u 2 u f u u 5 x r 3 u 2 x r 3 and e 2 u v x u 2 u g x u k x u 5 x r 3 u 2 x r 3 where 0 e 0 v k c r 3 r f c r r and g c r 3 r r are super linear at infinity instead of the expression sufficiently large in the existing works we give a certain range 0 under which the first system admits a ground state solution when v is positive and periodic under some mild conditions on v k and g we show that the second system has at least one nontrivial solution provided e 0 e 0 where the bound e 0 0 is expressed explicitly in terms of v k and g

borg s criterion for almost periodic differential equations borg s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction in this article this method is generalized to almost periodic differential equations both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution the condition uses a riemannian metric and an example for the explicit construction of such a metric is presented

existence and nonexistence of positive solutions for a static schrodinger poisson slater equation abstract in this paper we study the following type of the schrodinger poisson slater equation with critical growth u u 2 1 4 x u u p 1 u u 4 u in r 3 where 0 and p 11 7 5 for the case of p 2 5 we develop a novel perturbation approach together with the well known mountion pass theorem to prove the existence of positive ground states for the case of p 2 we obtain the nonexistence of nontrivial solutions by restricting the range of and also study the existence of positive solutions by the constrained minimization method for the case of p 11 7 2 we use a truncation technique developed by brezis and oswald 9 together with a measure representation concentration compactness principle due to lions 27 to prove the existence of radial symmetrical positive solutions for 0 with some 0 the above results nontrivially extend some theorems on the subcritical case obtained by ianni and ruiz 18 to the critical case

koszul and local cohomology and a question of dutta for a mathfrak m a local ring we study the natural map from the koszul cohomology module h dim a mathfrak m a to the local cohomology module h dim a mathfrak m a we prove that the injectivity of this map characterizes the cohen macaulay property of the ring a we also answer a question of dutta by constructing normal rings a for which this map is zero

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

two positive solutions of a class of schrodinger poisson system with indefinite nonlinearity abstract we study the existence and multiplicity of positive solutions of a class of schrodinger poisson system u u l x u k x u p 2 u h x u in r 3 l x u 2 in r 3 where 4 p 6 k c r 3 k changes sign in r 3 and lim x k x k 0 we mainly prove the existence of at least two positive solutions in the case that 1 and near 1 where 1 is the first eigenvalue of i d in h 1 r 3 with weight function h whose corresponding eigenfunction is denoted by e 1 an interesting phenomenon is that we do not need the condition r 3 k x e 1 p d x 0 which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity see e g s alama g tarantello on semilinear elliptic equations with indefinite nonlinearities calc var partial differential equations 1 1993 439 475

schrodinger poisson system with singular potential abstract consider the following schrodinger poisson system sp u v x u x u u p 1 u x y z r 2 r u 2 lim x x 0 where v 1 y with 0 y x 1 x 2 r 2 and y x 1 2 x 2 2 when 0 8 and max 2 2 2 p 5 the existence and a priori estimate of positive solutions of problem sp are established in suitable weighted sobolev space moreover the asymptotic behavior of the solutions as 0 is also discussed

existence of multiple solutions for modified schrdingerkirchhoffpoisson type systems via perturbation method with sign changing potential in this paper we prove the existence of positive solutions and negative solutions for the following modified schrdingerkirchhoffpoisson type systems a br3u2 u v x u u12u u2 f x u inr3 u2 inr3 where a 0 b0 v and f are continuous functions and v x is allowed to be sign changing under some certain assumptions on v and f we prove the existence of nontrivial nonnegative solutions nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method

uniqueness of finite energy solutions for maxwell dirac and maxwell klein gordon equations we prove uniqueness of solutions to the maxwell dirac system in the energy space namely we also give a proof for uniqueness of finite energy solutions to the maxwell klein gordon equations which is simpler than that given in 16

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

infinitely many standing wave solutions for the nonlinear klein gordon maxwell system with sign changing potential in this paper we study the infinitely many solutions for the nonlinear klein gordon maxwell system email protected v x u email protected email protected fu f x u email protected r 3 email protected email protected u 2 email protected r 3 where w 0 is a constant u f r 3 r the potential v x is allowed to be sign changing and the primitive of the nonlinearity f is of super linear growth near infinity our super linear conditions are weaker than the usual super linear conditions

existence and instability of standing waves with prescribed norm for a class of schrodinger poisson equations in this paper we study the existence and the instability of standing waves with prescribed l norm for a class of schrodinger poisson slater equations in r 0 1 i t x 1 p 2 0 when p 10 3 6 to obtain such solutions we look to critical points of the energy functional f u 1 2 ou 2l2 r3 1 4

a differential lyapunov framework for contraction analysis lyapunov s second theorem is an essential tool for stability analysis of differential equations the paper provides an analog theorem for incremental stability analysis by lifting the lyapunov function to the tangent bundle the lyapunov function endows the state space with a finsler structure incremental stability is inferred from infinitesimal contraction of the finsler metrics through integration along solutions curves

multi bump solutions for the nonlinear schrodinger poisson system in this paper we study a kind of nonlinear schrodinger poisson system with a parameter e for any positive integer m we prove that there exists e m 0 such that for 0 e e m the equation has an m bump positive solution under some suitable conditions as a consequence the equation has more and more multi bump positive solutions as e 0

ground state solutions of schrodinger poisson systems with variable potential and convolution nonlinearity abstract in the present paper we consider the following nonlinear schrodinger poisson system with convolution nonlinearity u v x u u i f u f u x r 3 u 2 x r 3 where 0 3 i r 3 r is the riesz potential v c r 3 0 f c r r and f t 0 t f s d s satisfies lim t f t t with min 2 6 4 by using some new analytic techniques and new inequalities we prove the above system admits a ground state solution under mild assumptions on v and f in particular our results cover and improve the existing ones for the schrodinger poisson system which can be considered as the limited problem when 0

multi bump solutions and multi tower solutions for equations on rn let 0 be a small parameter in this paper we study existence of multiple multi bump positive solutions for the semilinear schrodinger equation u u 1 a x u p 2u u h1 rn where n 1 2 2 if n 1 or n 2 a c rn a x 0 for x rn and lim x a x 0 we also study existence of multiple multi tower positive solutions for the prescribed scalar curvature equation u 1 k x un 2n 2 u d1 2 rn where n 3 k c 0 k r 0 for r 0 limr 0k r 0 and limr k r 0