on a jumping problem for quasilinear elliptic equations let us denote by k the eigenvalues of the operator with homogeneous dirichlet condition first studied in 1 when 1 2 this problem has been widely investigated in the case that some eigenvalue k belongs to the interval see e g 14 13 10 and references therein the methods used are often a combination of topological and variational techniques in this paper we will be concerned with a jumping problem associated to a quasilinear elliptic equation of the form n

existence and stability of standing waves for a planar gauged nonlinear schrodinger equation abstract in this article we prove the existence stability and mass collapse behavior of standing waves for the chern simons schrodinger model in the plane this is a covariant nonlinear schrodinger type problem which is mass critical and arises from the chern simons theory in the defocusing case 1 we prove the existence of stable standing waves with arbitrary mass however in the focusing case 1 stable standing waves with small mass exist and we give a mass collapse behavior of the standing waves

coupled replicator equations for the dynamics of learning in multiagent systems starting with a group of reinforcement learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems we show that although agents model their environment in a self interested way without sharing knowledge a game dynamics emerges naturally through environment mediated interactions an application to rock scissors paper game interactions shows that the collective learning dynamics exhibits a diversity of competitive and cooperative behaviors these include quasiperiodicity stable limit cycles intermittency and deterministic chaos behaviors that should be expected in heterogeneous multiagent systems described by the general replicator equations we derive

chern simons limit of the standing wave solutions for the schrodinger equations coupled with a neutral scalar field abstract we show the existence of standing wave solutions to the schrodinger equation coupled with a neutral scalar field we also verify the chern simons limit for these solutions more precisely we prove that solutions to eqs 1 3 1 4 converge to the unique positive radially symmetric solution of the nonlinear schrodinger equation 1 6 as the coupling constant q goes to infinity

on a free boundary problem for minimal surfaces forc 4 embedded manifoldss 3 which are differmorphic to the standard sphere in 3 the existence of non constant minimal surfaces bounded bys and intersectings orthogonally along their boundaries is deduced

existence and concentration of semiclassical ground state solutions for the generalized chern simons schrodinger system in h1 r2 abstract this paper is concerned with the following singularly perturbed problem in h 1 r 2 e 2 u v x u a 0 u x u j 1 2 a j 2 u x u f u e 1 a 2 u x 2 a 1 u x 1 2 u 2 1 a 1 u x 2 a 2 u x 0 e a 0 u 1 a 2 u 2 2 a 1 u 2 where e is a small parameter v c r 2 r and f c r r by using some new variational and analytic techniques joined with the manifold of pohozaev nehari type we prove that there exists a constant e 0 0 determined by v and f such that for e 0 e 0 the above problem admits a semiclassical ground state solution v e with exponential decay at infinity we also establish a new concentration behaviour of v e as e 0 in particular our results are available to the nonlinearity f u u s 2 u for s 4 6 which extend the existing results concerning the case f u u s 2 u for s 6

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

numerical implementation of dynamical mean field theory for disordered systems application to the lotka volterra model of ecosystems dynamical mean field theory dmft is a tool that allows one to analyze the stochastic dynamics of n interacting degrees of freedom in terms of a self consistent 1 body problem in this work focusing on models of ecosystems we present the derivation of dmft through the dynamical cavity method and we develop a method for solving it numerically our numerical procedure can be applied to a large variety of systems for which dmft holds we implement and test it for the generalized random lotka volterra model and show that complex dynamical regimes characterized by chaos and aging can be captured and studied by this framework

solutions of second order and fourth order odes on the half line abstract we start by studying the existence of positive solutions for the differential equation u a x u g u with u 0 u 0 where a is a positive function and g is a power or a bounded function in other words we are concerned with even positive homoclinics of the differential equation the main motivation is to check that some well known results concerning the existence of homoclinics for the autonomous case where a is constant are also true for the non autonomous equation this also motivates us to study the analogous fourth order boundary value problem u 4 c u a x u u p 1 u u 0 u 0 0 u u 0 for which we also find nontrivial and in some instances positive solutions

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

vortex condensations of a non relativistic maxwell chern simons theory abstract we study an elliptic system subject to periodic boundary conditions arising from the study of vortex condensates for a selfdual non relativistic model introduced by dunne et al phys rev d 43 1332 1345 1990 in the framework of chern simons gauge theory we prove the existence of a vortex condensate for all values of the relevant physical parameters involved in the model this completes a recent work of spruck and yang 16 who could deal only with a restricted set of those parameters uniqueness is also discussed

standing waves of nonlinear schrodinger equations with the gauge field abstract we study standing waves for nonlinear schrodinger equations with the gauge field some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field a in terms of complex scalar field we also show that there exists no standing wave for certain range of parameters by establishing a new inequality of sobolev type

multiple solutions for a class of nonlinear sturm liouville problems when nonlinearities are not odd abstract this article concerns a sturm liouville problem whose nonlinearity is not odd a variational method is used to obtain the existence of infinitely many solutions which are distinguished by their nodal properties however an example shows some nodal classes can have no solutions while others do

a numerical solution to the gel fand levitan marchenko equation in this paper a numerical method is developed to solve the inverse scattering problem associated with two gel fand levitan marchenko integral equations this inverse problem is represented by two coupled partial differential equations the solution of these partial differential equations is two kernel functions these kernels are also related to each other through the gel fand levitan marchenko integral equations legendre gauss lobatto nodes are used to construct the n t h polynominal interpolation to approximate the solution of the kernel functions which are related to the scattering potential an example is given to demonstrate the accuracy of the proposed method

multiple solutions for a class of nonlinear sturm liouville problems on the half line abstract a class of nonlinear sturm liouville problems is considered these problems admit zero as a trivial solution variational techniques and approximation arguments are used to obtain the existence of nontrivial solutions with any prescribed number of nodes results are also obtained in higher dimensions via investigation of the set of radial solutions of appropriate partial differential equations

infinitely many nonradial solutions of a euclidean scalar field equation abstract we study the equation u b x u x u x r n u h 1 r n the existence of a nonradial solution has been an open problem for some time even in the autonomous case under suitable hypotheses mainly on we find an unbounded sequence of nonradial solutions if n 4 or n 6 we also obtain infinitely many solutions if b and are not rotationally symmetric with respect to x but satisfy only a weaker symmetry

infinitely many solutions to fourth order superlinear periodic problems we present a new min max approach to the search of multiple t periodic solutions to a class of fourth order equations u iv t cu t f t u t t 0 t where f t u is continuous t periodic in t and satisfies a superlinearity assumption when u for every n n we prove the existence of a t periodic solution having exactly 2n zeroes in 0 t

on the chern simons limit for a maxwell chern simons model on bounded domains abstract this paper concerns the chern simons limit for the abelian maxwell chern simons model on bounded domains with vanishing gauge fields we prove that every sequence of solutions of the maxwell chern simons equations has a subsequence converging to a solution of the chern simons equation in any c k norms we also show that the maxwell chern simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star shaped domains

a positive solution of a nonlinear schrodinger equation with g symmetry abstract in this paper we consider the following nonlinear schrodinger equations u v x u f u in r n u h 1 r n here n 2 and v x c r n r is invariant under a group action g o n we deal with both finite group actions and infinite group actions the existence of a positive solution will be shown for a wide class of nonlinearities f u

algorithm for counting large directed loops we derive a belief propagation algorithm for counting large loops in a directed network we evaluate the distribution of the number of small loops in a directed random network with given degree sequence we apply the algorithm to a few characteristic directed networks of various network sizes and loop structures and compare the algorithm with exhaustive counting results when possible the algorithm is adequate in estimating loop counts for large directed networks and can be used to compare the loop structure of directed networks and their randomized counterparts

mass extinction in a dynamical system of evolution with variable dimension introducing the effect of extinction into the so called replicator equations in mathematical biology we construct a general model where the diversity of species i e the dimension of the equation is a time dependent variable the system shows very different behavior from the original replicator equation and leads to mass extinction when the system initially has high diversity the present theory can serve as a mathematical foundation for the paleontologic theory for mass extinction this extinction dynamics is a prototype of dynamical systems where the variable dimension is inevitable

solutions without any symmetry for semilinear elliptic problems abstract we prove the existence of infinitely many solitary waves for the nonlinear klein gordon or schrodinger equation u u u 3 0 in r 2 which have finite energy and whose maximal group of symmetry reduces to the identity

special measure function separability and reflectionless potentials a relationship is found between the separability of the spectral measure function and the poschl teller pt reflectionless potential rlp the limits on this relationship are briefly discussed

ambiguities and almost reflectionless potentials this paper considers possible ambiguities associated with perturbations on reflectionless solutions of the gel fand levitan equation by considering perturbations on the spectral measure function and the potential it is shown that no ambiguity exists with respect to spectral measure perturbations however it is shown that first order effects in the potential can lead to second order effects in the spectral measure function thereby demonstrating an ambiguity with respect to first order approximations

model ecosystems with random nonlinear interspecies interactions the principle of competitive exclusion in ecology establishes that two species living together cannot occupy the same ecological niche here we present a model ecosystem in which the species are described by a series of phenotypic characters and the strength of the competition between two species is given by a nondecreasing modulating function of the number of common characters using analytical tools of statistical mechanics we find that the ecosystem diversity defined as the fraction of species that coexist at equilibrium decreases as the complexity i e number of characters of the species increases regardless of the modulating function by considering both selective and random elimination of the links in the community web we show that ecosystems composed of simple species are more robust than those composed of complex species in addition we show that the puzzling result that there exists either rich or poor ecosystems for a linear modulating function is not typical of communities in which the interspecies interactions are determined by a complementarity rule

sign changing solutions to a gauged nonlinear schrodinger equation abstract we study the existence and asymptotic behavior of the least energy sign changing solutions to a gauged nonlinear schrodinger equation u u h 2 x x 2 x h s s u 2 s d s u u p 2 u x r 2 u h r 1 r 2 where 0 p 6 and h s 1 2 0 s r u 2 r d r combining constraint minimization method and quantitative deformation lemma we prove that the problem possesses at least one least energy sign changing solution u which changes sign exactly once moreover we show that the energy of u is strictly larger than two times of the ground state energy finally the asymptotic behavior of u as 0 is also analyzed

existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in this paper by variational methods we study the existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in particular the first result is based on the ambrosetti rabinowitz condition moreover the second results are established via the poho aev identity finally the last results are not based on the ambrosetti rabinowitz condition and we use generalized conditions to prove the results

existence and asymptotic behavior of nodal solutions for the kirchhoff type problems in r3 abstract in this paper we study the existence and asymptotic behavior of nodal solutions to the following kirchhoff problem a b r 3 u 2 d x u v x u f x u in r 3 u h 1 r 3 where v x is a smooth function a b are positive constants because the so called nonlocal term r 3 u 2 d x u is involved in the equation the variational functional of the equation has totally different properties from the case of b 0 under suitable construction conditions we prove that for any positive integer k the problem has a sign changing solution u k b which changes signs exactly k times moreover the energy of u k b is strictly increasing in k and for any sequence b n 0 n there is a subsequence b n s such that u k b n s converges in h 1 r 3 to w k as s where w k also changes signs exactly k times and solves the following equation a u v x u f x u in r 3 u h 1 r 3

random matrices and condensation into multiple states in the present work we employ methods from statistical mechanics of disordered systems to investigate static properties of condensation into multiple states in a general framework we aim at showing how typical properties of random interaction matrices play a vital role in manifesting the statistics of condensate states in particular an analytical expression for the fraction of condensate states in the thermodynamic limit is provided that confirms the result of the mean number of coexisting species in a random tournament game we also study the interplay between the condensation problem and zero sum games with correlated random payoff matrices

a method to construct reflectionless potentials this paper gives a method with an example of constructing a potential which is near reflectionless for some range of energies the assumed structure of the desired reflection coefficient r k is known function that is filtered by an inverted gaussian this gives an r k that is zero at particular energy and is exponentially near zero in a surrounding range

infinitely many radial and non radial solutions to a quasilinear schrodinger equation abstract in this paper we are concerned with the following quasilinear schrodinger equation u u u 2 v x u f x u x r n by using a change of variables we obtained the existence of a sequence of radial solutions for n 2 a sequence of nonradial solutions for n 4 or n 6 and a nonradial solution for n 5

asymptotics for maxwell chern simons multivortices chern simons gauge theories are of interest in several areas of physics including high tc superconductivity as they can describe particles which are both electrically and magnetically charged on the other hand the introduction of the chern simons term into the lagrangian results in serious di5culties for a rigorous treatment of the corresponding equations of motion therefore a considerable e7ort has been devoted in recent years to the study of models with a self dual structure such a structure allows us to obtain time independent energy minimizer solutions to the full second order equations of motion by solving appropriate rst order equations also known as the bogomol nyi equations 11 these particular solutions are called multivortices for an overview of self dual chern simons theories see the monograph of dunne 8 in 9 10 hong et al and jackiw and weinberg introduced a chern simons model where the maxwell term is neglected in order to achieve self duality we call this model the cs model the cs multivortices have proved to be particularly interesting from the mathematical viewpoint and have been extensively analyzed see references

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

complex dynamics in learning complicated games game theory is the standard tool used to model strategic interactions in evolutionary biology and social science traditionally game theory studies the equilibria of simple games however is this useful if the game is complicated and if not what is we define a complicated game as one with many possible moves and therefore many possible payoffs conditional on those moves we investigate two person games in which the players learn based on a type of reinforcement learning called experience weighted attraction ewa by generating games at random we characterize the learning dynamics under ewa and show that there are three clearly separated regimes i convergence to a unique fixed point ii a huge multiplicity of stable fixed points and iii chaotic behavior in case iii the dimension of the chaotic attractors can be very high implying that the learning dynamics are effectively random in the chaotic regime the total payoffs fluctuate intermittently showing bursts of rapid change punctuated by periods of quiescence with heavy tails similar to what is observed in fluid turbulence and financial markets our results suggest that at least for some learning algorithms there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory

ecological communities with lotka volterra dynamics ecological communities in heterogeneous environments assemble through the combined effect of species interaction and migration understanding the effect of these processes on the community properties is central to ecology here we study these processes for a single community subject to migration from a pool of species with population dynamics described by the generalized lotka volterra equations we derive exact results for the phase diagram describing the dynamical behaviors and for the diversity and species abundance distributions a phase transition is found from a phase where a unique globally attractive fixed point exists to a phase where multiple dynamical attractors exist leading to history dependent community properties the model is shown to possess a symmetry that also establishes a connection with other well known models

symmetric and nonsymmetric solutions for an elliptic equation on rn abstract we obtain for n 4 the existence of a positive solution as well as a nonradial solution which changes the sign of the equation u v z u f z u in r n u h 1 r n where the function f is a superlinear nonlinearity and the function v is a potential that is bounded below by a positive constant as a consequence we complete for the case v z 1 and f z u f u the well known work of bartsh and willen j funct anal 117 1993 261 because there it does not consider the case n 5

multiplicity of positive and nodal solutions for scalar field equations abstract in this paper the question of finding infinitely many solutions to the problem u a x u u p 2 u in r n u h 1 r n is considered when n 2 p 2 2 n n 2 and the potential a x is a positive function which is not required to enjoy symmetry properties assuming that a x satisfies a suitable slow decay at infinity condition and moreover that its graph has some dips we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions the proof method is purely variational and allows to describe the shape of the solutions

infinitely many global continua bifurcating from a single solution of an elliptic problem with concave convex nonlinearity abstract we study the bifurcation of solutions of semilinear elliptic boundary value problems of the form u f x u u in u 0 on on an annulus r n with a concave convex nonlinearity a special case being the nonlinearity first considered by ambrosetti brezis and cerami f x u u u q 2 u u p 2 u with 1 q 2 p although the trivial solution u 0 0 is nondegenerate if 0 we prove that 0 u 0 0 0 is a bifurcation point in fact the bifurcation scenario is very singular we show that there are infinitely many global continua of radial solutions c j r c 1 j n 0 that bifurcate from the trivial branch r 0 at 0 u 0 0 0 and consist of solutions having precisely j 1 nodal annuli a detailed study of these continua shows that they accumulate at r 0 0 so that every 0 with 0 is a bifurcation point moreover adding a point at infinity to c 1 they also accumulate at r so there is bifurcation from infinity at every r

group theoretical approach to reflectionless potentials we examine the general form of potentials with zero reflection coefficient in one dimensional hamiltonians connected with casimir invariants of non compact groups

existence theorems for periodic non relativistic maxwell chern simons solitons abstract we study the system of elliptic equations formula defined over a doubly periodic domain in r 2 where the coefficients are specifically given by the physical model this system arises in a self dual non relativistic maxwell chern simons theory coupled with a neutral scalar field in 2 1 dimensional spacetime and the solutions represent multivortices known as condensates our existence results reveal that the number of vortices confined in a periodic cell domain can be arbitrary and that the chern simons coupling parameter imposes no restriction to the existence of solutions

nodal standing waves for a gauged nonlinear schrodinger equation in r2 abstract this paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear schrodinger equation u u h 2 x x 2 x h s s u 2 s d s u u p 2 u x r 2 u x u x h 1 r 2 where 0 p 6 and h s 1 2 0 s r u 2 r d r is the so called chern simons term we prove that for any positive integer k the problem has a sign changing solution u k which changes sign exactly k times moreover the energy of u k is strictly increasing in k and for any sequence n n there exists a subsequence n s such that n s 1 p 2 u k n s converges in h 1 r 2 to w k as s where w k also changes sign exactly k times and solves the following equation u u u p 2 u u h 1 r 2

non topological multivortex solutions to the self dual maxwell chern simons higgs systems in this paper we construct non topological multivortex solutions to the non relativistic self dual maxwell chern simons higgs system in r2 which make the energy functional finite moreover our proof of the existence of solutions reveals precise asymptotic behavior of solutions near spatial infinity using exactly the same method we also establish the existence of non topological multivortex solutions to the relativistic self dual maxwell chern simons higgs system

multiplicity results for sign changing bound state solutions of a semilinear equation abstract in this paper we give conditions on f so that the problem u f u 0 x r n n 2 has at least two radial bound state solutions with any prescribed number of zeros and such that u 0 belongs to a specific subinterval of 0 this property will allow us to give conditions on f so that this problem has at least any given number of radial solutions having a prescribed number of zeros

random replicators with high order interactions we use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of n species interacting via random mutual interactions as well as via deterministic self interactions of order p 2 we show that the main effect of increasing the order of the interactions among the species is to make the system less competitive in the sense that the fraction of extinct species is greatly reduced in addition we find that for p 2 there is a threshold value which gives a lower bound to the concentration of the surviving species preventing then the existence of rare species and consequently increasing the robustness of the ecosystem to external perturbations

matrix games mixed strategies and statistical mechanics matrix games constitute a fundamental problem of game theory and describe a situation of two players with completely conflicting interests we show how methods from statistical mechanics can be used to investigate the statistical properties of optimal mixed strategies of large matrix games with random payoff matrices and derive analytical expressions for the value of the game and the distribution of strategy strengths in particular the fraction of pure strategies not contributing to the optimal mixed strategy of a player is calculated both independently distributed as well as correlated elements of the payoff matrix are considered and the results compared with numerical simulations

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

rank abundance relations in evolutionary dynamics of random replicators we present a nonequilibrium statistical mechanics description of rank abundance relations rar in random community models of ecology specifically we study a multispecies replicator system with quenched random interaction matrices we here consider symmetric interactions as well as asymmetric and antisymmetric cases rars are obtained analytically via a generating functional analysis describing fixed point states of the system in terms of a small set of order parameters and in dependence on the symmetry or otherwise of interactions and on the productivity of the community our work is an extension of tokita phys rev lett 93 178102 2004 where the case of symmetric interactions was considered within an equilibrium setup the species abundance distribution in our model come out as truncated normal distributions or transformations thereof and in some case are similar to left skewed distributions observed in ecology we also discuss the interaction structure of the resulting food web of stable species at stationarity cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher order interactions

complementarity and diversity in a soluble model ecosystem complementarity among species with different traits is one of the basic processes affecting biodiversity defined as the number of species in the ecosystem we present here a soluble model of microbial based ecosystems in which the species are characterized by binary traits and their pairwise interactions follow a complementarity principle manipulation of the species composition and so the study of its effects on the species diversity is achieved through the introduction of a bias parameter favoring one of the traits using statistical mechanics tools we find explicit expressions for the allowed values of the equilibrium species concentrations in terms of the control parameters of the model

ecological communities from random generalized lotka volterra dynamics with nonlinear feedback we investigate the outcome of generalized lotka volterra dynamics of ecological communities with random interaction coefficients and nonlinear feedback we show in simulations that the saturation of nonlinear feedback stabilizes the dynamics this is confirmed in an analytical generating functional approach to generalized lotka volterra equations with piecewise linear saturating response for such systems we are able to derive self consistent relations governing the stable fixed point phase and to carry out a linear stability analysis to predict the onset of unstable behavior we investigate in detail the combined effects of the mean variance and covariance of the random interaction coefficients and the saturation value of the nonlinear response we find that stability and diversity increases with the introduction of nonlinear feedback where decreasing the saturation value has a similar effect to decreasing the covariance we also find cooperation to no longer have a detrimental effect on stability with nonlinear feedback and the order parameters mean abundance and diversity to be less dependent on the symmetry of interactions with stronger saturation

standing waves for a schrodinger chern simons higgs system abstract we consider a system arising from a nonrelativistic chern simon higgs model in which a charged field is coupled with a gauge field we prove an existence result for small coupling constants

approximate solution to the gel fand levitan equation an approximate solution to the gel fand levitan equation is given for the case where g r s can be approximated by an n t h rank separable function it is necessary both to show that k r s is n t h rank of g r s and to introduce a new orthogonal set

on the multiplicity of radial solutions to superlinear dirichlet problems in bounded domains in this paper we are concerned with the existence and multiplicity of nodal solutions to the dirichlet problem associated to the elliptic equation u q x g u 0 in a ball or in an annulus in rn the nonlinearity g has a superlinear and subcritical growth at infinity while the weight function q is nonnegative in 0 1 and strictly positive in some interval r1 r2 0 1 by means of a shooting approach together with a phase plane analysis we are able to prove the existence of infinitely many solutions with prescribed nodal properties

sign changing solutions for a class of kirchhoff type problem in bounded domains abstract we are interested in the existence of least energy sign changing solutions for a class of kirchhoff type problem in bounded domains because the so called nonlocal term b u 2 d x u is involving in the equation the variational functional of the equation has totally different properties from the case of b 0 combining constraint variational method and quantitative deformation lemma we prove that the problem possesses one least energy sign changing solution u b moreover we show that the energy of u b is strictly larger than the ground state energy finally we regard b as a parameter and give a convergence property of u b as b 0

random replicators with asymmetric couplings systems of interacting random replicators are studied using generating functional techniques while replica analyses of such models are limited to systems with symmetric couplings dynamical approaches as presented here allow us specifically to address cases with asymmetric interactions where there is no lyapunov function governing the dynamics we focus on replicator models with gaussian couplings of general symmetry between p 2 species and discuss how an effective description of the dynamics can be derived in terms of a single species process upon making a fixed point ansatz persistent order parameters in the ergodic stationary states can be extracted from this process and different types of phase transitions can be identified and related to each other we discuss the effects of asymmetry in the couplings on the order parameters and the phase behaviour for p 2 and p 3 numerical simulations verify our theory for the case of cubic interactions numerical experiments indicate regimes in which only a finite number of species survives even when the thermodynamic limit is considered

nodal soliton solutions for quasilinear schrodinger equations with critical exponent this paper is concerned with constructing nodal radial solutions for quasilinear schrodinger equations in rn with critical growth which have appeared as several models in mathematical physics for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem since the critical exponent appears and the lower order term may change sign we should use more delicate arguments

radial solutions of dirichlet problems with concave convex nonlinearities abstract we prove the existence of a double infinite sequence of radial solutions for a dirichlet concave convex problem associated with an elliptic equation in a ball of r n we are interested in relaxing the classical positivity condition on the weights by allowing the weights to vanish the idea is to develop a topological method and to use the concept of rotation number the solutions are characterized by their nodal properties

infinitely many sign changing solutions for a schrodinger equation in rn abstract in this paper we consider a schrodinger equation u a x 1 u f u applying principle of symmetric criticality and the invariant set method under some assumptions on a and f we obtain an unbounded sequence of radial sign changing solutions for the above equation in r n when 0 large enough as n 4 or n 6 0 given using fountain theorem and the principle of symmetric criticality we prove that there exists an unbounded sequence of non radial sign changing solutions for the above equation in r n

standing waves for the chern simons schrodinger equation with critical exponential growth abstract in this paper by combing the variational methods and trudinger moser inequality we study the existence and multiplicity of the positive standing waves for the following chern simons schrodinger equation 0 1 u u 0 h s s u 2 s d s h 2 x x 2 u f x u k x in r 2 where h s 0 s l 2 u 2 l d l 0 and the nonlinearity f r 2 r r behaves like exp u 2 as u for the case 0 we can get a mountain pass type solution

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

standing waves of the schrodinger equation coupled with the chern simons gauge field we prove the existence of infinitely many radially symmetric standing wave solutions of the chern simons schrodinger system our result is established by applying the mountain pass theorem to the functional which is obtained by representing gauge fields a in terms of a scalar field

radial solutions of superlinear equations on ℝn part i global variational approach we present a global variational approach to the search for multiple nodal solutions u h 1 n to a class of elliptic equations of type u x f x u x x n where n 2 f is superlinear and subcritical and f x 0

infinitely many radial and non radial solutions for a fractional schrodinger equation in this paper we study the following fractional schrodinger equation s u v x u f x u x r n where s s 0 1 denotes the fractional laplacian by variational methods we obtain the existence of a sequence of radial solutions for n 2 a sequence of non radial solutions for n 4 or n 6 and a non radial solution for n 5

broken ergodicity and memory in the minority game we study the dynamics of the batch minority game with market impact correction using generating functional techniques to carry out the quenched disorder average we find that the assumption of weak long term memory which one usually makes in order to calculate ergodic stationary states breaks down when the persistent autocorrelation becomes larger than cc0 772 we show that this condition remarkably coincides with the at line found in an earlier static calculation this result suggests a new scenario for ergodicity breaking in disordered systems

nonself dual chern simons and maxwell chern simons vortices on bounded domains abstract we first show the existence of the multivortex solutions of maxwell chern simons higgs mcsh model on bounded domains and prove that the solutions of the euler lagrange equations of the mcsh energy functional converge to those of the abelian higgs ah model and the chern simons higgs csh model in suitable limits respectively we also show the existence of the multivortex solutions of the nonself dual csh model on bounded domains besides we study asymptotics for the minimizers of mcsh energy functional when the gauge field vanishes