on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in rn abstract in this paper we study the elliptic system of gradient type with nonstandard growth conditions in r n existence and multiplicity results under two growth conditions on the reaction terms are established

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

locally distinguishing inputs for systems on lie groups and parallelism this paper is concerned with invariant systems on lie groups a sufficient condition for a piecewise constant input to distinguish between any two distinct initial states contained in some open neighbourhood of the state space is given this condition involves the parallel translation of vectors along curves thus relating observability to parallelism

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

on superlinear p x laplacian equations in rn abstract we consider the p x laplacian equations in r n the nonlinearity is superlinear but does not satisfy the ambrosetti rabinowitz type condition we obtain ground states of the equations improving a recent result of fan x l fan p x laplacian equations in r n with periodic data and nonperiodic perturbations j math anal appl 341 2008 103 119 we also establish a bartsch wang type compact embedding theorem for variable exponent spaces then a multiplicity result for the equations is proved for odd nonlinearity

existence of radial solutions for p x laplacian equations in rn abstract in this paper we obtain sufficient conditions of the existence of radial solutions for p x laplacian equations in r n

existence and multiplicity of solutions for a neumann problem involving the p x laplace operator abstract in this paper we study the following nonlinear neumann boundary value problem div u p x 2 u u p x 2 u f x u x t r u v 0 x t r where r n is a bounded domain with smooth boundary u v is the outer unit normal derivative on 0 is a real number p is a continuous function on with inf x p x 1 f r r is a continuous function using the three critical point theorem due to ricceri under the appropriate assumptions on f we establish the existence of at least three solutions of this problem some known results are generalized

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

the principle of concentration compactness in lp x spaces and its application abstract in this paper we establish a principle of concentration compactness in l p x spaces and apply it to obtain the existence of solutions for p x laplacian equations with critical growth

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

harnack s inequality for p harmonic functions with unbounded exponent p abstract we study properties of the function u lim u where u is the solution of the min p laplacian dirichlet problem with bounded sobolev boundary function here p n is a variable exponent such that 1 p is lipschitz continuous we derive bloch type estimates and using them we prove harnack s inequality in cases of unbounded but finite exponent

on quantum state observability and measurement we consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable the system is assumed to be a finite dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a lie group we investigate to what extent by an appropriate sequence of evolutions and measurements we can obtain information on the initial state of the system we present a system theoretic viewpoint of this problem in that we study the observability of the system in this context we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification

existence of strong solutions of a p x laplacian dirichlet problem without the ambrosetti rabinowitz condition abstract in this paper we consider the existence of strong solutions of the following p x laplacian dirichlet problem via critical point theory d i v u p x 2 u f x u in u 0 on we give a new growth condition under which we use a new method to check the cerami compactness condition hence we prove the existence of strong solutions of the problem as above without the growth condition of the well known ambrosetti rabinowitz type and also give some results about multiplicity of the solutions

on the sub supersolution method for p x laplacian equations abstract this paper deals with the sub supersolution method for the p x laplacian equations a sub supersolution principle for the dirichlet problems involving the p x laplacian is established it is proved that the local minimizers in the c 1 topology are also local minimizers in the w 1 p x topology for given energy functionals a strong comparison theorem for the p x laplacian equations is presented some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p x laplacian equations are given

the local boundedness and harnack inequality of p x laplace equation using moser s iteration method we investigate the local boundedness of solutions for p x laplacian equations and derive a harnack type inequality important for applications

superlinear problems without ambrosetti and rabinowitz growth condition superlinear elliptic boundary value problems without ambrosetti and rabinowitz growth condition are considered existence of nontrivial solution result is established by combining some arguments used by struwe and tarantello and schechter and zou also by wang and wei firstly by using the mountain pass theorem due to ambrosetti and rabinowitz is constructed a solution for almost every parameter by varying the parameter then it is considered the continuation of the solutions

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

elliptic equations and systems with nonstandard growth conditions existence uniqueness and localization properties of solutions abstract we study the dirichlet problem for the elliptic equations i d i a i x u d i u p i x 2 d i u c x u u x 2 u f x in a bounded domain r n and the class of elliptic systems j d j a i j x u f i x u i 1 n u u 1 u n satisfying the growth condition x s v r n 2 i j a i j x v v i j a 0 i j v i j p i j x a 0 const 0 the exponents p i j x p i x x are known functions these equations are usually referred to as elliptic equations with nonstandard growth conditions we prove first the theorems of existence of bounded weak solutions and establish sufficient conditions of uniqueness of a weak solution our main purpose is the study of the localization properties of weak solutions we show that the weak solution may identically vanish on a set of nonzero measure in a dead core and derive estimates on the size and location of these dead cores in terms of the problem data the study of the localization properties is performed via the method of local energy estimates

transience of edge reinforced random walk we show transience of the edge reinforced random walk errw for small reinforcement in dimension d ge3 this proves the existence of a phase transition between recurrent and transient behavior thus solving an open problem stated by diaconis in 1986 the argument adapts the proof of quasi diffusive behavior of the supersymmetric susy hyperbolic model for fixed conductances by disertori et al commun math phys 300 435 486 2010 using the representation of errw as a mixture of vertex reinforced jump processes vrjp with independent gamma conductances and the interpretation of the limit law of vrjp as a susy hyperbolic sigma model developed by sabot and tarres j eur math soc arxiv 1111 3991 2015

on local observability for invariant systems on lie groups and coset spaces the purpose of this short paper is to provide a proof of a local observability criterion for invariant systems on lie groups and coset spaces the criterion was originally presented with an ad hoc proof based on lie theoretic arguments the alternative proof presented here shows that the criterion essentially comes from the classical observability results involving the observability codistribution and its properties in this way it unifies the two approaches and extends the application field of the classical results

phase transition for the once reinforced random walk on mathbb z d like trees in this short paper we consider the once reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk on each of these trees we prove that there exists an explicit critical parameter a 0 such that the once reinforced random walk is almost surely recurrent if a a 0 and almost surely transient if a a 0 this provides the first examples of phase transition for the once reinforced random walk

a strong maximum principle for differential equations with nonstandard p x growth conditions abstract in this paper with some special technics we give a strong maximum principle for the equations with nonstandard p x growth conditions div x u u d x f x u 0 in where x s d x f x u satisfy some conditions

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

the existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p laplacian type without the ambrosetti rabinowitz condition abstract in this paper we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p laplacian type p p u f x u x u 0 x where p 1 r 1 r n is a bounded domain and p u d i v u p 2 u is the p laplacian of u f c 0 r 1 r 1 is p superlinear at t 0 and subcritical at t we prove that under suitable conditions for all 0 the problem p has at least one nontrivial solution without assuming the ambrosetti rabinowitz condition our main result extends a result for p for when p 2 given by miyagaki and souto 2008 in 8 to the general problem p where p 1 meanwhile our result is stronger than a similar result of li and zhou 2003 given in 15

interpolation theorems for variable exponent lebesgue spaces when hardy littlewood maximal operator is bounded on lp rn space we prove lp rn bmo rn lq rn where q p 1 and lp rn h1 rn lq rn where 1 q 1 p

on superlinear problems without the ambrosetti and rabinowitz condition abstract existence and multiplicity results are obtained for superlinear p laplacian equations without the ambrosetti and rabinowitz condition to overcome the difficulty that the palais smale sequences of the euler lagrange functional may be unbounded we consider the cerami sequences our results extend the recent results of miyagaki and souto o miyagaki m souto superlinear problems without ambrosetti and rabinowitz growth condition j differential equations 245 2008 3628 3638

existence and asymptotic behavior of positive solutions for variable exponent elliptic systems abstract we mainly consider the existence and asymptotic behavior of positive solutions of the following system p x u p x f v in p x v p x g u in u v 0 on where r n is a bounded domain with c 2 boundary 1 p x c 1 is a function p x u div u p x 2 u is called p x laplacian we proved the existence of a positive solution when is large enough and lim u f m g u 1 p 1 u p 1 0 for every m 0 where p inf x p x in particular we do not assume any symmetric condition and we do not assume any sign condition on f 0 and g 0

solutions for p x laplacian dirichlet problems with singular coefficients using the critical point theory we investigate the existence and multiplicity of solutions for p x laplacian dirichlet problem with singular coefficients

picone identities for half linear elliptic operators with p x laplacians and applications to sturmian comparison theory picone identities are established for a class of half linear elliptic operators with p x laplacians and sturmian comparison theorems are obtained on the basis of the picone identities specializations to half linear partial or ordinary differential inequalities with p x laplacians are shown

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

p x laplacian equations satisfying cerami condition the goal of this paper is to prove the existence and multiplicity of solutions for the p x laplacian equations without the well known ambrosetti rabinowitz type growth conditions the critical point theorems with cerami condition are used

on the transience of processes defined on galton watson trees we introduce a simple technique for proving the transience of certain processes defined on the random tree mathcal g generated by a supercritical branching process we prove the transience for once reinforced random walks on mathcal g that is a generalization of a result of durrett kesten and limic probab theory related fields 122 2002 567 592 moreover we give a new proof for the transience of a family of biased random walks defined on mathcal g other proofs of this fact can be found in ann probab 16 1988 1229 1241 and ann probab 18 1990 931 958 as part of more general results a similar technique is applied to a vertex reinforced jump process a by product of our result is that this process is transient on the 3 ary tree davis and volkov probab theory related fields 128 2004 42 62 proved that a vertex reinforced jump process defined on the b ary tree is transient if b 4 and recurrent if b 1 the case b 2 is still open

once edge reinforced random walk on a tree we consider a nearest neighbor walk on a regular tree with transition probabilities proportional to weights or conductances of the edges initially all edges have weight 1 and the weight of an edge is increased to c 1 when the edge is traversed for the first time after such a change the weight of an edge stays at c forever we show that such a walk is transient for all values of c ge 1 and that the walk moves off to infinity at a linear rate we also prove an invariance principle for the height of the walk

observability of systems on lie groups and coset spaces the purpose of this paper is to study the observability of a class of systems for which the state space is a lie group and the output space is a cosec space the study of this type of system was initiated by brockett siam j control 10 1972 pp 265 284 in this paper brockett s observability results are generalized and necessary and sufficient conditions for observability are obtained effective algorithms are established to verify such conditions finally as an application some disturbance decoupling problems are considered

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

on the boundary blow up solutions of p x laplacian equations with singular coefficient abstract this paper investigates the problem p x u x f x u 0 in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian x is a singular coefficient the existence and nonexistence of boundary blow up solutions are discussed and the singularity of the boundary blow up solution is given

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated