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

positive and negative solutions for a class of kirchhoff type problems on unbounded domain abstract in this paper we study the existence of multiple positive and negative solutions for a class of kirchhoff type problems on an unbounded domain using the variational methods we show that our problem admits at least two positive and two negative solutions under different conditions a cut off functional is utilized to obtain the bounded palais smale sequences

existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

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 of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

infinitely many solutions for a fractional kirchhoff type problem via fountain theorem abstract in this paper we use the fountain theorem and the dual fountain theorem to study the existence of infinitely many solutions for kirchhoff type equations involving nonlocal integro differential operators with homogeneous dirichlet boundary conditions a model for these operators is given by the fractional laplacian of kirchhoff type m r 2 n u x u y 2 x y n 2 s d x d y s u x u f x u in u 0 in r n where is a smooth bounded domain of r n s is the fractional laplacian operator with 0 s 1 and 2 s n is a real parameter m is a continuous and positive function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

on a schrodinger kirchhoff type equation involving the p x laplacian abstract in this paper we deal with a schrodinger type equation involving a nonlocal kirchhoff type coefficient and depending on two real parameters working within the framework of variable exponent spaces and using the variational approach we obtain several results of existence and multiplicity of solutions depending on the range of the parameters

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

global bifurcation and positive solution for a class of fully nonlinear problems in this paper we study global bifurcation phenomena for the following kirchhoff type problem m u x 2 d x u f x u in u 0 on where m is a continuous function under some natural hypotheses we show that 1 a m 0 0 is a bifurcation point and there is a global continuum c emanating from 1 a m 0 0 where 1 a denotes the first eigenvalue of the above problem with f x s a x s as an application of the above result we study the existence of positive solution for this problem with asymptotically linear nonlinearity

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

existence of nontrivial solutions for kirchhoff type variational inclusion system in r n abstract in this paper we consider the following kirchhoff type variational inclusion system a b r n u 2 dx u v x u 1 f u v in r n c d r n v 2 dx v v x v 2 f u v in r n with u v h 1 r n where n 2 and f r 2 r is a locally lipschitz function and i f u v i 1 2 are the partial generalized gradients in the sense of clarke under proper growth conditions on the nonlinearity f the existence of nontrivial solutions for the above system is established our approach is variational based on the theories of nonsmooth analysis

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

multiplicity results for solutions of p biharmonic problems abstract in this paper we deal with the multiplicity existence of solutions for the following p biharmonic equation p 2 u u p 2 u u q 2 u in u u 0 on where is a bounded domain in r n p 2 u u p 2 u p q p n p n 2 p r is a parameter when p q p we prove that the above problem possesses infinitely many solutions while when q p a multiplicity existence result is obtained

fractional kirchhoff type equation with hardy littlewood sobolev critical exponent abstract in this paper we consider the following fractional kirchhoff type equation a b r n r n u x u y 2 x y n 2 s d x d y s u i u 2 h u 2 h 2 u in r n where n 3 s 0 1 a 0 b 0 0 n and 2 h n n 2 s is the hardy littlewood sobolev critical exponent and i is the riesz potential we establish the existence nonexistence and multiplicity of nontrivial solutions to above equation

the hartogs extension theorem on n 1 complete complex spaces employing morse theory for the global control of monodromy and the method of analytic discs for local extension we establish a version of the global hartogs extension theorem in a singular setting for every domain d of an n 1 complete normal complex space x of pure dimension n 2 and for every compact set k in d such that d k is connected holomorphic or meromorphic functions in d k extend holomorphically or meromorphically to d normality is an unvavoidable assumption for holomorphic extension but we show that meromorphic extension holds on a reduced globally irreducible not necessarily normal x of pure dimension n 2 provided that the regular part of d k is connected

multiplicity of solutions for variable order fractional kirchhoff equations with nonstandard growth abstract the aim of this paper is to study a degenerate kirchhoff type elliptic problem driven by the fractional laplace operator with variable order derivative and variable exponents more precisely we consider u s 2 1 s u a x u p x 2 u b x u q x 2 u in u 0 in r n where u s is the garliado seminorm 1 n 1 s r n r n 0 1 is a continuous function 0 is a parameter is a bounded domain in r n with n 2 s x y for all x y s is the variable order fractional laplacian a b l are two positive weight functions and p q c with 1 p x 2 q x 2 n n 2 s x x under some suitable assumptions we obtain that the problem admits at least two distinct nonnegative solutions by applying the nehari manifold approach provided is sufficiently small moreover the existence of infinitely many solutions is also investigated by the symmetric mountain pass theorem

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

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

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

infinitely many non negative solutions for a p x kirchhoff type problem with dirichlet boundary condition abstract in this paper we consider the dirichlet problem involving the p x kirchhoff type a b 1 p x u p x d x 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

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

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem

multiple solutions of an even order nonlinear problem with convex concave nonlinearity abstract we study multiple solutions of an even order nonlinear partial differential equation with a dirichlet boundary condition a related class of nonlinear systems is investigated

an elliptic generalization of schur s pfaffian identity we present a pfaffian identity involving elliptic functions whose rational limit gives a generalization of schur s pfaffian identity for pf xj xi xj xi this identity can be regarded as the pfaffian counterpart of frobenius generalization of cauchy s determinant identity for det 1 xi xj