sobolev type embedding and quasilinear elliptic equations with radial potentials abstract we study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials the existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions

multiplicity of solutions for singular quasilinear schrodinger equations with critical exponents abstract by variational methods we study the existence of multiplicity of solutions for singular quasilinear schrodinger equations

multiple and sign changing solutions for a class of semilinear biharmonic equation abstract in this paper under an improved hardy rellich s inequality we study the existence of multiple and sign changing solutions for a biharmonic equation in unbounded domain by the minimax method and linking theorem

on an elliptic problem with critical exponent and hardy potential in this paper we study the following elliptic problem with critical exponent and a hardy potential u x 2u u u 2 2u u h01 where is a smooth open bounded domain in rn n 3 which contains the origin and 2 is the critical sobolev exponent we show that if n 5 and 0 n 22 2 n 2n 2 this problem has a ground state solution for each fixed 0 moreover we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions if n 7 and 0 n 22 2 4 this problem has infinitely many sign changing solutions for each fixed 0

generating hyperbolical rotation matrix for a given hyperboloid abstract hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by a 1 x 2 a 2 y 2 a 3 z 2 r in this paper we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the lorentzian scalar product space r a 1 a 2 a 3 2 1 such that the general hyperboloids are the pseudo spheres of r a 1 a 2 a 3 2 1 we adapt the rodrigues cayley and householder methods to r a 1 a 2 a 3 2 1 and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix

a new hyperchaotic system from the lu system and its control this paper presents a 4d new hyperchaotic system which is constructed by a linear controller to a 3d lu system some complex dynamical behaviors such as hopf bifurcation chaos and hyperchaos of the simple 4d autonomous system are investigated and analyzed the corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories lyapunove exponents bifurcation path analysis of power spectrum and poincare projections furthermore the design is illustrated with both simulations and experiments finally the control problem of a new hyperchaotic system is investigated using negative feedback control ordinary feedback control dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium numerical simulations are presented to demonstrate the effectiveness of the proposed controllers

existence results for schrodinger p laplace equations involving critical growth in rn abstract we establish some existence results for schrodinger p laplace equations in r n with various potentials and critical growth of nonlinearity that may occur on some nonempty set although not necessarily the whole space r n the proofs are mainly based on concentration compactness principles in a suitable weighted variable exponent sobolev space and its imbeddings

symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents this paper deals with a class of singular biharmonic elliptic systems involving critical exponents in a bounded symmetric domain by using the variational method and the symmetric criticality principle of palais we obtain several existence and multiplicity results of g symmetric solutions for the systems

higher order singular problems of caffarelli kohn nirenberg lin type abstract we prove the existence of nontrivial critical points of the functional j u r n 1 p x a k u p h x x a k u p 1 q q x x b u q d x related to the caffarelli kohn nirenberg inequality and its higher order variant by lin as a consequence we obtain nontrivial solutions of the degenerate elliptic equation x a p u p 2 u h x x a k p u p 2 u q x x b q u q 2 u we also show that when p 2 j has infinitely many critical points

multiplicity of solutions for homogeneous elliptic systems with critical growth abstract in this paper we are concerned with the number of nonnegative solutions of the elliptic system u q u u v 1 2 h u u v in v q v u v 1 2 h v u v in u v 0 on where r n is a bounded smooth domain n 4 2 2 n n 2 and q u h u and q v h v are the partial derivatives of the homogeneous functions q h c 1 r 2 r where r 2 0 0 in the proofs we apply variational methods and ljusternik schnirelmann theory

existence and multiplicity of positive solutions to elliptic systems involving critical exponents abstract in this paper a singular elliptic system involving multiple critical exponents and the caffarelli kohn nirenberg inequality is investigated by using the extremals of the best hardy sobolev constants the existence and multiplicity of positive solutions to the system are established

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

realizing hybrid synchronization of time delay hyperchaotic 4d systems via partial variables hybrid synchronization of time delay hyperchaotic 4d systems with different initial conditions is investigated via partial variables based on the lyapunov stability theory sufficient conditions are given to get the hybrid synchronization by constructing a novel lyapunov function in the form of exponential form it is proved that some partial state variables of two time delay hyperchaotic systems are anti synchronized while other state variables are complete synchronized when one or two linear controllers are imposed on the response system simultaneously the unknown parameters in the response system are also estimated completely numerical simulation results are presented to demonstrate the effectiveness of the proposed chaos synchronization scheme

multiplicity of self similar solutions for a critical equation we consider the equation u 12 x u f u u 2 2u x rn with 0 f superlinear and 2 2n n 2 for n 3 we prove that for each k n there exists k 0 such that the equation has at least k pairs of solutions provided 0 in the proof we use variational methods for the even functional associated to the equation

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions in this paper the existence and multiplicity results of solutions for the following quasilinear elliptic problem 0 1 div x ap u p 2 u h x u p 2u g x u r 2u x x ap u p 2 u f x u q 2u x are established where is an exterior domain in rn with the compact and smooth boundary by the variational methods we prove that the problem 0 1 has at least two positive solutions at the last part of the paper we also consider the critical case and prove the existence of solution by concentration compactness principle

on the symmetric scalar curvature problem on sn abstract we discuss some existence results dealing with the scalar curvature problem on s n in the presence of various symmetries

the decomposition of an orthogonal transformation as a product of reflections in this work an algorithm to decompose a given orthogonal transformation as a product of reflections through hyperplanes is presented this in fact constitutes a constructive proof of a cartan theorem valid over any field k q r or c clifford algebras are used to explicitly calculate the reflections that decompose a given orthogonal transformation our algorithm may have application in fields such as computer graphics or crystallography and can also play an important role in orthogonal elimination and the solution of equations systems an explicit example is provided and we apply our results to the crystallographic problem of coincidence lattices

anti synchronization of uncertain unified chaotic systems with dead zone nonlinearity abstract this paper addresses chaos anti synchronization of uncertain unified chaotic systems with dead zone input nonlinearity using the sliding mode control technique and lyapunov stability theory a proportional integral pi switching surface is proposed to ensure the stability of the closed loop error system in sliding mode then a sliding mode controller smc is proposed to guarantee the hitting of the switching surface even with uncertainties and the control input containing dead zone nonlinearity some simulation results are included to demonstrate the effectiveness and feasibility of the proposed synchronization scheme

chaotic bursting lag synchronization of hindmarsh rose system via a single controller chaotic bursting lag synchronization of hindmarsh rose system is investigated two lag synchronization schemes with only a single controller are proposed to synchronize hindmarsh rose chaotic system via back stepping method especially in the second scheme only one state variable is contained in the controller based on lyapunov stability theory the sufficient conditions for synchronization are obtained analytically in both cases finally numerical simulations are provided to show the effectiveness of the developed methods

time scale synchronization of chaotic oscillators this paper presents the result of the investigation of chaotic oscillator synchronization a new approach to detect the synchronized behaviour of chaotic oscillators has been proposed this approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators this approach has been applied for the coupled rossler and lorenz systems

g reflectors analogues of householder transformations in scalar product spaces we characterize the analogues of householder transformations in matrix groups associated with scalar products and precisely delimit their mapping capabilities given a matrix group image and vectors x y necessary and sufficient conditions are derived for the existence of a householder like analogue image such that gx y when g exists we show how it can be constructed from x and y examples of matrix groups to which these results apply include the symplectic and pseudo unitary groups

constructive ways for generating generalized real orthogonal matrices as products of generalized symmetries the cartan dieudonne scherk cds theorem of algebraic group theory asserts that for fields of characteristic different from 2 every generalized orthogonal matrix can be written as the product of a certain minimal number of generalized symmetries when computing over r or c these symmetries are called householder matrices in numerical linear algebra they play an important role in orthogonal elimination and the qr factorization of matrices this note gives constructive proofs of a number of results to generate a generalized real or complex orthogonal or unitary matrix as the product of generalized householder matrices the proofs are patterned after the standard householder elimination practises of numerical analysis the original proofs of the cds theorem of around 50 years ago were not constructive but relied on the nonexistence of certain isotropic subspaces with respect to the underlying generalized inner product

elliptic systems involving the critical exponents and potentials abstract in this paper we investigate a singular elliptic system which involves the critical sobolev exponent and multiple hardy type terms by employing variational methods the existence of its positive solutions is established by the moser iteration method some asymptotic properties of its nontrivial solutions at the singular points are verified

a concentration compactness lemma with applications to singular eigenvalue problems let omega subset of or equal to r n be any open set we study the nonlinear eigenvalue problem delta p u lambda v x u p 2 u u epsilon d 0 1 p omega where 1 p n and v is an element of l loc 1 omega may have strong singularities and an indefinite sign the key ingredient is a precised concentration compactness lemma related to v dependent limiting problems this work follows extends and simplifies that of a tertikas 1998 j funct anal 154 42 66 dealing with the positive linear case for omega r n c 1999 academic press

adaptive anti synchronization of chaotic systems with fully unknown parameters this paper centers on the chaos anti synchronization between two identical or different chaotic systems using adaptive control the sufficient conditions for achieving the anti synchronization of two chaotic systems are derived based on lyapunov stability theory an adaptive control law and a parameter update rule for unknown parameters are introduced such that the chen system is controlled to be the lorenz system theoretical analysis and numerical simulations are shown to verify the results

multiplicity of solutions for a noncooperative p laplacian elliptic system in rn in this paper we study a system of p laplacian elliptic equations pu u p 2u fu x u v inrn pv v p 2v fv x u v inrn u v w1 p rn applying principle of symmetric criticality and a limit index theory under some assumptions on fu fv we can obtain the existence of infinitely many radial solutions and nonradial solutions for the above system which extends some results in li nonlinear anal tma 25 1995 1371

on the singular elliptic systems involving multiple critical sobolev exponents in this paper a singular elliptic system is investigated which involves multiple critical sobolev exponents and hardy type terms by using variational methods and analytical techniques the existence of positive and sign changing solutions to the system is established

existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical hardy sobolev exponents abstract this paper deals with the singular semilinear elliptic problem div x 2 a u u x 2 1 a q x u p 2 u x b p h x u in u 0 on where r n n 3 is a smooth bounded domain 0 and is g symmetric with respect to a subgroup g of o n 0 a n 2 2 0 0 with n 2 2 a 2 2 a b a 1 p p a b 2 n n 2 1 a b q x is continuous and g symmetric on and h r r is a continuous nonlinearity of lower order satisfying some conditions based upon the symmetric criticality principle of palais and variational methods we prove several existence and multiplicity results of g symmetric solutions under certain appropriate hypotheses on q and h

nodal solutions for p laplace equations with critical growth abstract this paper is concerned with the existence and nodal character of the nontrivial solutions for the following quasilinear elliptic equations involving critical sobolev exponents 1 i 1 n x i u p 2 u x i u p 2 u u p 2 u f u u w r 1 p r n where p 2 and p np n p is the critical sobolev exponent for the embedding w r 1 p r n l p r n the function f satisfies some conditions given by f1 f2 f3 in the paper the main results obtained in this paper are that there exists at least a pair of nontrivial solutions for eq 1 provided that n p2 and there exists at least a pair of nontrivial solutions uk uk of eq 1 for each k n 0 such that both uk and uk possess exactly k nodes provided that f t 0 for t 0 and n p2 p 1 p

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

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

existence results for kirchhoff equations with hardy littlewood sobolev critical nonlinearity abstract in this paper we study a class of kirchhoff equations with hardy littlewood sobolev critical nonlinearity in r n more precisely we consider a b r n u 2 d x u a u 2 u i u p u p 2 u i u 2 2 u 2 2 2 u where a 0 b 0 n 3 0 4 n 3 n 4 2 n n p 2 2 n n 2 0 and i is a riesz potential we prove the existence and multiplicity of solutions for the equation by variational methods together with concentration compactness principle

a general scheme for q s synchronization of chaotic systems abstract this paper addresses the q s synchronization between chaotic and or hyper chaotic systems based on the lyapunov stability theorem a general scheme for q s synchronization of chaotic and or hyper chaotic systems is proposed the drive and response systems discussed in this paper can be strictly different dynamical systems including different dimensional systems four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed general scheme for q s synchronization

positive solutions to a fourth order elliptic problem by the lusternik schnirelmann category abstract in this paper we consider the fourth order problem 2 u u s 1 u u 2 2 u in u u 0 in u u 0 on where is a smooth bounded domain in r n n 5 and 2 2 n n 4 we assume 2 s 1 2 in case n 8 and 2 2 s 1 2 for the critical dimensions n 5 6 7 then we prove that if has a rich topology described by its lusternik schnirelmann category then the problem has multiple solutions at least as many as cat in case the parameter 0 is sufficiently small

chaotic phase synchronization and phase diffusion critical behaviors after the breakdown of chaotic phase synchronization in coupled two chaotic oscillators are studied by observing phase diffusion for phase difference of the oscillators particularly by paying attention to similarity and difference of the critical behavior of the angular velocity

an algorithm for the cartan dieudonne theorem on generalized scalar product spaces abstract we present an algorithmic proof of the theorem on generalized real scalar product spaces with arbitrary signature we use clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes the relationship with the cartan dieudonne scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation

existence and nonexistence of maximizers for variational problems associated with trudinger moser type inequalities in mathbb r n we discuss the existence of a maximizer for a maximizing problem associated with the trudinger moser type inequality in mathbb r n n geq 2 different from the bounded domain case we obtain both of the existence and the nonexistence results the proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences

existence and multiplicity of symmetric solutions for the weighted critical quasilinear problems this paper deals with the weighted critical quasilinear problem div u p 2 u x a p m u p 2u x p a 1 q x u p a b 2u x b p a b h x u in w u 0on w where w r n is a smooth bounded domain 0 w and w is g symmetric with respect to a subgroup g of o n 1

on symmetric solutions of a singular elliptic equation with critical sobolev hardy exponent this paper is concerned with the existence of the nontrivial solutions of the following problem u u x 2 k x u2 s 1 x s x rn u dg1 2 rn where n 2 k x is a bounded continuous function satisfying some conditions dg1 2 rn is an appropriate sobolev space of g symmetric functions 2 s 2 n s n 2 is the critical sobolev hardy exponent and 0 s 2 0 n 22 2

existence and multiplicity of solutions to a singular elliptic system with critical sobolev hardy exponents and concave convex nonlinearities abstract this paper is concerned with a singular elliptic system which involves the caffarelli kohn nirenberg inequality and critical sobolev hardy exponents the existence and multiplicity results of positive solutions are obtained by variational methods

multiple solutions for a class of biharmonic elliptic systems with sobolev critical exponent abstract in this paper we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic systems with sobolev critical exponent in a bounded domain by using the variational method and the nehari manifold we obtain the existence and multiplicity results of nontrivial solutions for the systems

solutions for a class of quasilinear choquard equations with hardy littlewood sobolev critical nonlinearity abstract in this article we consider the quasilinear choquard equation with critical nonlinearity in r n e 2 u v x u e 2 u u 2 r n u 2 2 x y d y u 2 2 2 u h x u u where 0 n n 3 2 2 n n 2 is the critical exponent in the sense of hardy littlewood sobolev inequality e is a real parameter by using a change of variables the quasilinear equations are reduced to a semilinear one whose associated functionals are well defined in the usual sobolev space and satisfy the geometric conditions of the mountain pass theorem under suitable assumptions on v and h we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory in order to overcome the lack of compactness we apply the concentration compactness principle