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

boundary trace embedding theorems for variable exponent sobolev spaces we study boundary trace embedding theorems for variable exponent sobolev space w1 p let be an open bounded or unbounded domain in rn satisfying strong local lipschitz condition under the hypotheses that p l 1 infp x supp x n we prove that there is a continuous boundary trace embedding w1 p lq provided q a measurable function on satisfies condition p x q x n 1 p x n p x for x

bivariate c1 cubic spline spaces with homogeneous boundary conditions over fvs triangulation in this paper we mainly generalize the results in h w liu d hong d q cao bivariate c 1 cubic spline space over a nonuniform type 2 triangulation and its subspaces with boundary conditions comput math appl 49 2005 1853 1865 from the type 2 triangulation to the so called fvs triangulation a triangulated quadrangulation we study the bivariate c 1 cubic spline spaces s 3 1 0 email protected and s 3 1 1 email protected with homogeneous boundary conditions over an fvs triangulation email protected the dimensions are obtained and the locally supported bases are constructed for these spline spaces furthermore we also study the explicit bezier ordinates of the interpolation basis splines on a representative triangulated quadrilateral the results of this paper can be applied in many fields such as the finite element method for partial differential equation computer aided geometric design numerical approximation and so on

on variational inequalities with maximal monotone operators and multivalued perturbing terms in sobolev spaces with variable exponents abstract we are concerned in this paper with variational inequalities of the form a u v u f u v u l v u v k u k where a is a maximal monotone operator f is an integral multivalued lower order term and k is a closed convex set in a sobolev space of variable exponent we study both coercive and noncoercive inequalities in the noncoercive case a sub supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub and supersolutions

on the existence of weak solutions of nonlinear degenerate parabolic system with variable exponents abstract the main goal of the present paper is establishing the existence and uniqueness of weak solutions for the nonlinear degenerate reaction diffusion system with variable exponents a model also is proposed to characterize the invasion of cancer cells towards healthy cells with acidification environment moreover the main results of this paper are obtained using regularization problem the faedo galerkin approximation method some apriori estimates compactness results and the gronwall lemma

infinitely many non negative solutions for a dirichlet problem involving p x laplacian abstract in this paper we consider a dirichlet problem involving the p x laplacian of the type div u p x 2 u f x u in u 0 on we prove the existence of infinitely many non negative solutions of the problem by applying a general variational principle due to b ricceri and the theory of the variable exponent sobolev spaces

renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and l1 data abstract in this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and l 1 data and moreover we obtain the equivalence of renormalized solutions and entropy solutions

convergence relation between p x harmonic maps and minimizers of p x energy functional with penalization abstract we prove that the p x ginzburg landau type minimizers converge to the p x harmonic maps the location of zeros of the minimizers is presented in the case of p x 1 2 when p x 2 there exists no zero of the minimizers in the domain in addition the convergence rate of the modulus of the minimizers to 1 is estimated

elliptic boundary value problems with nonstandard growth conditions abstract we study different boundary value problems to the p x laplacean equation namely of robin and signorini types we prove the existence and uniqueness of weak solutions to the problems under study and also their continuous dependence on the exponent data

existence of infinitely many solutions for a neumann problem involving the p x laplacian abstract in this paper we consider the neumann problem involving the p x laplacian of the type div u p x 2 u x u p x 2 u f x u g x u in u 0 on we prove the existence of infinitely many solutions of the problem under weaker hypotheses by applying a variational principle due to b ricceri and the theory of the variable exponent sobolev spaces our results are an improvement and generalization of the relative results obtained by b ricceri for the p laplacian case

lp x ω estimates of vector fields and some applications to magnetostatics problems in the present paper we investigate holder type norm inequalities in terms of div and curl of the vector valued functions in variable exponent lebesgue spaces lp x where r3 moreover by using the obtained results we give some applications for magnetostatics problems

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

infinitely many solutions for a neumann type differential inclusion problem involving the p x laplacian abstract in this paper we consider a differential inclusion problem involving the p x laplacian of the type div u p x 2 u x u p x 2 u f x u g x u in u 0 on we prove the existence of infinitely many solutions of this problem under suitable hypotheses by applying a non smooth ricceri type variational principle and the theory of the variable exponent sobolev spaces

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

infinitely many homoclinic solutions for nonautonomous p t laplacian hamiltonian systems by using the symmetric mountain pass theorem we establish some existence criteria which guarantee that the second order ordinary p t laplacian systems of the form ddt email protected t p t email protected t a t u t p t 2u t email protected w t u t 0 have infinitely many homoclinic solutions where email protected r email protected r n email protected c r r and p t 1 email protected c r r and email protected c 1 rxr n r are non periodic in t

existence and multiplicity results for a new p x kirchhoff problem abstract in this work we study the existence and multiplicity results for the following nonlocal p x kirchhoff problem 0 1 a b 1 p x u p x d x d i v u p x 2 u u p x 2 u g x u in u 0 on where a b 0 are constants r n is a bounded smooth domain p c with n p x 1 is a real parameter and g is a continuous function the analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties

existence and multiplicity of solutions for dirichlet problem of p x laplacian type without the ambrosetti rabinowitz condition abstract this paper deals with the following variational problem with variable exponents div 1 u p x 1 u 2 p x u p x 2 u f x u in u 0 on where r n n 2 be a bounded domain with the smooth boundary p r is a lipschitz continuous function with 1 p ess inf x p x p x p ess sup x p x n and f c r r is superlinear but does not satisfy the usual ambrosetti rabinowitz type condition under three different superlinear conditions on f at infinity we prove that the above equation has at least a nontrivial solution moreover the existence of infinite many solutions is proved for odd nonlinearity especially some new tricks are introduced to show the boundedness of cerami sequences

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

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

on behaviour of solutions for a nonlinear viscoelastic equation with variable exponent nonlinearities abstract this paper deals with a viscoelastic wave equation with variable exponent nonlinearities subject to nonlinear boundary feedback under appropriate conditions a general decay result associated to solution energy is proved it is also shown that regarding arbitrary positive initial energy solutions blow up in a finite time

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 of positive solutions for elliptic systems with nonstandard p x growth conditions via sub supersolution method abstract we mainly consider the system p x u f v in p x v g u in u v 0 on where p x c 1 r n is a radial symmetric function which satisfies sup p x and 1 inf p x sup p x where p x u div u p x 2 u is called p x laplacian b 0 r r n is a bounded radial symmetric domain where r is sufficiently large we prove the existence of a positive solution when lim u f m g u 1 p 1 u p 1 0 in particular we do not assume any sign condition on f 0 and g 0

remarks on ricceri s variational principle and applications to the p x laplacian equations abstract in this paper we give some remarks on a variational principle of ricceri in the case of sequentially weakly lower semi continuous functionals defined on a reflexive real banach space in particular we introduce the notions of ricceri block and ricceri box which are more convenient in some applications than the weakly connected components using the variational principle of ricceri and a local mountain pass lemma we study the multiplicity of solutions of the p x laplacian equations with neumann dirichlet or no flux boundary condition and under appropriate hypotheses in which the integral functionals need not satisfy the ps condition on the global space we prove that the problem has at least seven solutions

global attractors for nonlinear parabolic equations with nonstandard growth and irregular data abstract we investigate the large time behavior of solutions to the following nonlinear parabolic equations u t div u p x 2 u f x u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 we first provide the existence and uniqueness of an entropy solution for the problem then through some delicate analysis we establish some regularity results on the solution by which we prove the existence of a global attractor for the solution semigroup

a compact embedding result for anisotropic sobolev spaces associated to a strip like domain and some applications abstract let m 1 and d 2 be integers and consider a strip like domain o r d where o r m is a bounded euclidean domain with smooth boundary furthermore let p o r d r be a uniformly continuous and cylindrically symmetric function we prove that the subspace of w 1 p x y o r d consisting of the cylindrically symmetric functions is compactly embedded into l o r d provided that m d p inf x y o r d p x y p sup x y o r d p x y as an application we study a neumann problem involving the p x y laplacian operator and an oscillating nonlinearity proving the existence of infinitely many cylindrically symmetric weak solutions our approach is based on variational and topological methods in addition to the principle of symmetric criticality

approximation of nonlinear wave equations with nonstandard anisotropic growth conditions weak solutions for nonlinear wave equations involving the p x laplacian for p 1 are constructed as appropriate limits of solutions of an implicit finite element discretization of the problem a simple fixed point scheme with appropriate stopping criteria is proposed to conclude convergence for all discretization regularization perturbation and stopping parameters tending to zero computational experiments are included to motivate interesting dynamics such as blowup and asymptotic decay behavior

stability of eigenvalues for variable exponent problems abstract in the framework of variable exponent sobolev spaces we prove that the variational eigenvalues defined by inf sup procedures of rayleigh ratios for the luxemburg norms are all stable under uniform convergence of the exponents

classification and geometrical properties of the xθ valued function spaces abstract this paper is devoted to investigate the x valued function spaces based on the notions of bounded topological lattice banach space net continuous modular net and the dual space net we divide x valued function spaces into two classes norm modular spaces and modular modular spaces for the first class we study the separability of l p i x give a representation of the dual space l p i x find sufficient conditions of the equality l p i x l p i x and prove the reflexivity of l p i x under some reasonable conditions and for the second class we prove the completeness and uniform convexity of l i x in suitable situations to show the naturality and rationality of these results some concrete function spaces such as l p t i l p x t l p t i w 1 p x t and l p t i w 0 1 p x t together with l p p t i l p x t are taken into account

a global attractor for a p x laplacian parabolic problem abstract in this work we study the asymptotic behavior of a p x laplacian parabolic problem of the form u t p x u b u in a bounded smooth domain in r n under dirichlet boundary conditions we prove under suitable assumptions the existence of a compact invariant global b attractor in l 2

quasi linear variable exponent boundary value problems with wentzell robin and wentzell boundary conditions let p c0 1 be such that 1 p p 1 p p let rn rn be a bounded w1 p w1 p extension domain and let be an upper d ahlfors measure supported on with d n p n d n p n we investigate the solvability of a class of quasi linear boundary value problems involving the p p laplace and p p laplace beltrami operators and either classical wentzell robin boundary conditions p ud u p 2 u dhn 1 u p 2ud 0on or general fully wentzell type boundary conditions of the form p ud p ud u p 2 u dhn 1 u p 2ud 0on where l d l d is such that infx x 0 for some constant 0 0 0 0 and u dhn 1 denotes the generalized p p normal derivative on in the interpretative sense we prove that the realization of the p p laplace operator with both of the above boundary conditions generate nonlinear ultracontractive submarkovian c0c0 semigroups on l2 dx l2 d l2 dx l2 d and hence their associated first order cauchy problems are both well posed on lq dx lq d lq dx lq d for all measurable function q with 1 q q 1 q q in addition we investigate the associated quasi linear elliptic problem with general wentzell boundary conditions and obtain existence uniqueness and global regularity of weak solutions to this equation

arbitrarily many solutions for a perturbed p x laplacian equation involving oscillatory terms this paper deals with a perturbed p x laplacian equation involving oscillatory terms div u p x email protected u u p x 2u q x f u email protected u email protected r n u 0 u x 0as x by using variational methods and the non smooth version symmetric criticality principle we establish a the unperturbed problem p 0 has infinitely many distinct solutions b the number of distinct solutions for p e becomes greater and greater whenever e is smaller and smaller our results are a generalization of the case of laplacian from kristaly to the case of p x laplacian

entire solutions of multivalued nonlinear schrodinger equations in sobolev spaces with variable exponent abstract we establish the existence of an entire solution for a class of stationary schrodinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows up at infinity the abstract framework is related to lebesgue sobolev spaces with variable exponent the proof is based on the critical point theory in the sense of clarke and we apply chang s version of the mountain pass lemma without the palais smale condition for locally lipschitz functionals our result generalizes in a nonsmooth framework a result of rabinowitz p h rabinowitz on a class of nonlinear schrodinger equations z angew math phys zamp 43 1992 270 291 on the existence of ground state solutions of the nonlinear schrodinger equation

on the p x laplacian robin eigenvalue problem abstract consider robin eigenvalue problem involving the p x laplacian on a smooth bounded domain as follows p x u u p x 2 u in u p x 2 u x u p x 2 u 0 on we prove the existence of infinitely many eigenvalue sequences if p x constant and also present some sufficient conditions for which there is no principal eigenvalue and the set of all eigenvalues is not closed

existence and asymptotic behavior of blow up solutions to a class of p x laplacian problems abstract this paper investigates the problem p x u f x u 0 in u x as d x 0 where p x u div u p x 2 u is called p x laplacian the existence of blow up solutions is discussed and the singularity of blow up solutions is given

global gradient estimates for the p laplacian abstract we consider calderon zygmund type estimates for the non homogeneous p laplacian system div d u p 2 d u div g p 2 g where p is a variable exponent we show that g p l q r n l 1 r n implies d u p l q r n l 1 r n for any q 1 we also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis the paper is an extension of the local estimates of acerbi mingione 2005

solutions for neumann boundary value problems involving p x laplace operators abstract in this paper we study the nonlinear neumann boundary value problem of the following form p div u p x 2 u u p x 2 u f x u in u p x 2 u g x u on using the variational method under appropriate assumptions on f and g we obtain a number of results on existence and multiplicity of solutions also what is also interesting is that we obtain some results which can be considered as extensions of the classical result named combined effects of concave and convex nonlinearities moreover the positive solution of the problem is considered

a gamma convergence approach to the critical sobolev embedding in variable exponent spaces abstract in this paper we study the critical sobolev embeddings w 1 p l p for variable exponent sobolev spaces from the point of view of the convergence more precisely we determine the limit of subcritical approximation of the best constant associated with this embedding as an application we provide a sufficient condition for the existence of extremals for the best constant

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

multiple solutions for a kirchhoff type problem involving the p x laplacian operator abstract the aim of this paper is to obtain weak solutions for a kirchhoff type problem involving the p x laplacian operator the main tool used for obtaining our result is a recent three critical points theorem established by ricceri

global gradient estimates for the parabolic p x t laplacian equation abstract in this paper we consider the global gradient estimates for weak solutions of the parabolic p x t laplacian type equation in a reifenberg flat domain

nonlinear damped wave equation in this paper we consider the following nonlinear wave equation with variable exponents uttu aut ut m 2 bu u p 2 where a b are positive constants by using the faedogalerkin method the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p we also prove the finite time blow up of solutions and give a two dimension numerical example to illustrate the blow up result

lipschitz type smoothness of the fractional integral on variable exponent spaces fil ramseyer mauricio javier consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet santa fe instituto de matematica aplicada del litoral universidad nacional del litoral instituto de matematica aplicada del litoral argentina

long time behavior for a nonlinear parabolic problem with variable exponents abstract this paper addresses the question of the asymptotic behavior of solutions to the p x laplacian problem u t div u p x 2 u f x u g with general assumptions on f x u and the exponent p x we prove the existence of global attractors in proper spaces then we consider the fractal dimension of global attractors for the problem under suitable conditions we show that the problem admits an infinite dimensional global attractor

mountain pass and ekeland s principle for eigenvalue problem with variable exponent in this article we study the boundary value problem div u p x 2 u u x 2 u u q x 2 u in u 0 on where is a smooth bounded domain in n and p q are continuous functions on we show that for any 0 there exists infinitely many weak solutions respectively if 0 and small enough then there exists a non negative non trivial weak solution our approach relies on the variable exponent theory of generalized lebesgue sobolev spaces combined with a 2 symmetric version for even functionals of the mountain pass theorem respectively on simple variational arguments based on ekeland s variational principle this article is devoted to the special issue vaxjo conference 2008

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

an eigenvalue problem with variable exponents abstract a highly nonlinear eigenvalue problem is studied in a sobolev space with variable exponent the euler lagrange equation for the minimization of a rayleigh quotient of two luxemburg norms is derived the asymptotic case with a variable infinity is treated local uniqueness is proved for the viscosity solutions

higher differentiability of minimizers of variational integrals with variable exponents in this paper we study the regularity properties of local minimizers of integral functionals of the form f v f x dv x dx where the integrand f is convex and satisfies the so called p x growth conditions

maximum principles for anisotropic elliptic inequalities we establish some maximum and comparison principles for weak distributional solutions of anisotropic elliptic inequalities in divergence form both in the homogeneous and non homogeneous case the main prototypes we have in mind are inequalities involving the p laplace operator and the generalized mean curvature operator

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

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems

convergence acceleration by extraction of linear subsequences logarithmic sequences are usually difficult to accelerate in this paper we show how to extract a subsequence converging linearly this subsequence can be easily accelerated by classical algorithms theoretical results are given as well as an algorithmic extraction procedure numerical examples illustrate the results

multiple solutions for noncooperative p x laplacian equations in rn involving the critical exponent abstract in this paper we study the multiplicity of solutions for a class of noncooperative p x laplacian equations in r n involving the critical exponent by using the limit index theory due to li y q li a limit index theory and its application nonlinear anal 25 1995 1371 1389 and concentration compactness principle we obtain a sequence of radially symmetric solutions

multiple solutions for nonlocal system of p x q x kirchhoff type abstract in this paper we study the existence of solutions for a neumann elliptic system with variable exponents under some suitable conditions and by applying an equivalent variational approach to a recent ricceri s three critical points theorem we established the existence of at least three weak solutions

the fibering map approach to a nonlocal problem involving p x laplacian abstract we study the existence of positive solutions for a p x kirchhoff problem the main tool used is the fibering map approach for the corresponding nehari manifold

existence and some properties of solutions for degenerate elliptic equations with exponent variable abstract in this paper we study degenerate elliptic equations with variable exponents when a perturbation term satisfies the ambrosetti rabinowitz condition and does not satisfy the ambrosetti rabinowitz condition for the first case we employ the standard mountain pass theorem to give the existence of solutions for the second case we use browder s theorem for monotone operators to show the unique existence of a solution when the perturbation term is decreasing with respect to a function variable a priori bound and nonnegativeness of solutions are also given we emphasize that the log holder continuous condition is not required

asymptotic stability of solutions to quasilinear hyperbolic equations with variable sources abstract in this paper we consider the following quasilinear hyperbolic equation involving variable sources u t t d i v u p x 2 u u t m x 2 u t u q x 2 u some energy estimates and k o m o r n i k inequality are used to prove some uniform estimates of decay rates of the solution and then we prove that u x t 0 is asymptotic stable in terms of natural energy associated with the solution of the above equation as we know such results are seldom seen for the variable exponent case at last we give some numerical examples to illustrate our results

uniform convexity of musielak orlicz sobolev spaces and applications abstract this paper deals with uniform convexity of musielak orlicz sobolev spaces and its applications to variational problems some sufficient conditions and examples for uniform convexity of musielak orlicz sobolev spaces are given some special properties relative to the uniformly convex modular for uniformly convex musielak orlicz sobolev spaces are presented as an application of these abstract results the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p x growth are studied

on the existence and stability of solutions for dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div a x u x p x 2 u x b x u x p x 2 u x f u k x u x u x 0 u w 0 1 p x with nonlinearity satisfying some local growth conditions we construct a new duality theory which differs from the known ones in that it does not require a type of a palais smale condition

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

on a dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div v z 1 x u v z n x u l u x u f u k x u u w 0 1 p x in a bounded domain and with nonconvex nonlinearity satisfying some local growth conditions the conditions upon v and l allow for considering the p x laplacian equation we use the relations between critical points and critical values to the primal and a suitable dual action functional to get the existence stability and some properties of the solutions

robin fractional problems with symmetric variable growth in this paper we study the fractional p laplacian and we introduce the corresponding nonlocal conormal derivative for this operator we prove the basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators in the second part of this paper see sec iv we prove the existence of weak solutions of corresponding p robin boundary problems with sign changing potentials by applying variational tools

variable exponent morrey and campanato spaces abstract the classical morrey spaces and campanato spaces are generalized to the variable exponent case the definitions and some basic properties of the variable exponent morrey and campanato spaces are presented a concept of the p average of the functions is introduced

a quasilinear neumann problem involving the p x laplacian abstract we study the following neumann problem p x u x u p x 2 u x f u g x u in u 0 on and we prove that under suitable assumptions on the functions f p and g the ricceri two local minima theorem together with the palais smale property ensures the existence of at least three solutions of it this work could be considered a possible extension of some results by cammaroto chinni and di bella who handled the case where p x is constant

multivariate spline space over cross cut partition in this paper we mainly study the generator bases of the prime module determined by the global conformality condition of multivariate spline over star cross cut partition the results can help us to get the dimension and construct the spline bases for multivariate spline space over cross cut partition quasi cross cut partition and super cross cut partition

three solutions for a neumann type differential inclusion problem involving the p x laplacian abstract in this paper we consider neumann type differential inclusion problems involving the p x laplacian of the type div u p x 2 u u p x 2 u f x u in u 0 on applying a version of the non smooth three critical points theorem we obtain the existence of three solutions of the problem in w 0 1 p x

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

sobolev s inequality for riesz potentials with variable exponent satisfying a log holder condition at infinity abstract our aim in this paper is to deal with the boundedness of maximal functions in generalized lebesgue spaces l p when p satisfies a log holder condition at infinity that is weaker than that of cruz uribe fiorenza and neugebauer d cruz uribe a fiorenza c j neugebauer the maximal function on variable l p spaces ann acad sci fenn math 28 2003 223 238 29 2004 247 249 our result extends the recent work of diening l diening maximal functions on generalized l p spaces math inequal appl 7 2004 245 254 and the authors futamura and mizuta t futamura y mizuta sobolev embeddings for riesz potential space of variable exponent preprint as an application of the boundedness of maximal functions we show sobolev s inequality for riesz potentials with variable exponent