ground state of kirchhoff type fractional schrodinger equations with critical growth abstract in this paper we study the critical kirchhoff type fractional schrodinger equation 0 1 1 r 3 r 3 u x u y 2 x y 3 2 s d x d y s u u f u u 2 s 1 in r 3 where s 0 1 and 2 s 6 3 2 s we establish the pohoz ev type identity of 0 1 when s 3 4 1 under some conditions on and f u we obtain some results on the existence of ground state solutions when s 0 3 4 we also prove the non existence result in particular when 0 we obtain an existence result

on a fractional schrodinger equation with periodic potential abstract in this paper we first consider the spectrum of the fractional schrodinger operator s v x on r n where s 0 1 and v x be continuous periodic in x using a new nonlocal normal derivative we prove that the operator has purely continuous spectrum which is bounded below and consists of closed disjoint intervals then when 0 belongs to a spectral gap of s v x we establish an existence result for the fractional schrodinger equation via a new linking theorem it is worth to mention that the nonlinear term in our problem does not satisfy the periodicity and the existence result is even new for the case s 1

shear correction factors and an energy consistent beam theory abstract presented here is a new derivation of shear correction factors for isotropic beams by matching the exact shear stress resultants and shear strain energy with those of the equivalent first order shear deformation theory moreover a new method of deriving in plane and shear warping functions from available elasticity solutions is shown the derived exact warping functions can be used to check the accuracy of a two dimensional sectional finite element analysis of central solutions the physical meaning of a shear correction factor is shown to be the ratio of the geometric average to the energy average of the transverse shear strain on a cross section examples are shown for circular and rectangular cross sections and the obtained shear correction factors are compared with those of cowper 1966 the energy averaged shear representative is also used to derive timoshenkos beam theory

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

existence and symmetry of solutions for critical fractional schrodinger equations with bounded potentials abstract this paper is concerned with the following fractional schrodinger equations involving critical exponents u v x u k x f u u 2 2 u in r n where is the fractional laplacian operator with 0 1 n 2 is a positive real parameter and 2 2 n n 2 is the critical sobolev exponent v x and k x are positive and bounded functions satisfying some extra hypotheses based on the principle of concentration compactness in the fractional sobolev space and the minimax arguments we obtain the existence of a nontrivial radially symmetric weak solution for the above mentioned equations without assuming the ambrosetti rabinowitz condition on the subcritical nonlinearity

existence of ground state solutions of nehari pohozaev type for fractional schrodinger poisson systems with a general potential abstract in this paper we consider the following fractional schrodinger poissonproblem s u v x u u f u in r 3 t u 2 in r 3 where 0 s t 1 and 2 s 2 t 3 the potential v x is weakly differentiable and f c r r by introducing some new tricks we prove that the problem admits a ground state solution of nehari pohozaev type under mild assumptions on v and f the results here extend the existing study

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

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

a non classical mindlin plate model incorporating microstructure surface energy and foundation effects a non classical model for a mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory a surface elasticity theory and a two parameter winkler pasternak foundation model it includes all five kinematic variables possible for a mindlin plate the equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on hamilton s principle and the microstructure surface energy and foundation effects are treated in a unified manner the newly developed model contains one material length scale parameter to describe the microstructure effect three surface elastic constants to account for the surface energy effect and two foundation parameters to capture the foundation effect the current non classical plate model reduces to its classical elasticity based counterpart when the microstructure surface energy and foundation effects are all suppressed in addition the new model includes the mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases recovers the kirchhoff plate model incorporating the microstructure surface energy and foundation effects and degenerates to the timoshenko beam model including the microstructure effect to illustrate the new mindlin plate model the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived

on the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity abstract in this paper we consider the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity e 2 s m u s a e 2 a e s u v x u u 2 s 2 u h x u 2 u x r n u x 0 as x where a e s is the fractional magnetic operator with 0 s 1 2 s 2 n n 2 s m r 0 r is a continuous nondecreasing function v r n r 0 and a r n r n are the electric and magnetic potentials respectively by using the fractional version of the concentration compactness principle and variational methods we show that the above problem i has at least one solution provided that e e and ii for any m n has m pairs of solutions if e e m where e and e m are sufficiently small positive numbers moreover these solutions u e 0 as e 0

short time critical dynamics and universality on a two dimensional triangular lattice abstract the critical scaling and universality in the short time dynamics for spin models on a two dimensional triangular lattice are investigated by using a monte carlo simulation emphasis is placed on the dynamic evolutions from fully ordered initial states to show that universal scaling exists already in the short time regime in forms of power law behavior of the magnetizations and binder cumulant the results estimated for the dynamic and static critical exponents z and confirm explicitly that the potts models on the triangular lattice and square lattice belong to the same universality classes our critical scaling analysis strongly suggests that the simulation for the dynamic relaxations can be used to determine the universality

a fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three dimensional stress effects abstract a new approach is used to develop a geometrically exact nonlinear beam model for naturally curved and twisted solid composite rotor blades undergoing large vibrations in three dimensional space a combination of the new concepts of local displacements and local engineering stresses and strains a new interpretation and manipulation of the virtual local rotations an exact coordinate transformation and the extended hamilton principle is used to derive six fully nonlinear equations of motion describing one extension two bending one torsion and two shearing vibrations of composite beams the formulation is based on an energy approach but the derivation is fully correlated with the newtonian approach and provides a straightforward explanation of all nonlinear structural terms without using complex tensor operations or asymptotic expansions the theory accounts for in plane and out of plane warpings due to bending extensional shearing and torsional loadings elastic couplings among warpings and three dimensional stress effects by using the results of a two dimensional static sectional finite element analysis also the theory fully accounts for extensionality initial curvatures and geometric nonlinearities the equations display linear elastic couplings due to structural anisotropy and initial curvatures and nonlinear geometric couplings the theory contains most of the existing beam theories as special cases and the final equations of motion are put in compact matrix form

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

finite size scaling and critical exponents in critical relaxation we simulate the critical relaxation process of the two dimensional ising model with the initial state both completely disordered or completely ordered results of a new method to measure both the dynamic and static critical exponents are reported based on the finite size scaling for the dynamics at the early time from the time dependent binder cumulant the dynamical exponent z is extracted independently while the static exponents beta nu and nu are obtained from the time evolution of the magnetization and its higher moments

on fractional schrodinger equation in rn with critical growth in this paper we study the following nonlinear fractional schrodinger equation with critical exponent h2 u v x u u 2 2u u q 2u x rn where h is a small positive parameter 0 2 0 is a parameter the potential v rn r is a positive continuous function satisfying some natural assumptions by using variational methods we obtain the existence of solutions in the following case if 2 0 such that for all 0 we show that it has one nontrivial solution and there exist at least cat nontrivial solutions if max 2 4 n 2 0

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

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

critical fractional p laplacian problems with possibly vanishing potentials abstract we obtain nontrivial solutions of a critical fractional p laplacian equation in the whole space and with possibly vanishing potentials in addition to the usual difficulty of the lack of compactness associated with problems involving critical sobolev exponents the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments we overcome this difficulty using a generalized linking construction based on the z 2 cohomological index

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

existence of solutions for asymptotically periodic fractional schrodinger equation abstract in this paper we study the following nonlinear fractional schrodinger equation s u v x u f x u x r n where s 0 1 s is the fractional laplacian operator of order s and v f are asymptotically periodic functions in x under suitable conditions the existence of nontrivial solution is obtained by using variational methods and concentration compactness argument

couple stress based strain gradient theory for elasticity the deformation behavior of materials in the micron scale has been experimentally shown to be size dependent in the absence of stretch and dilatation gradients the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures in the couple stress theory formulation only conventional equilibrium relations of forces and moments of forces are used the couple s association with position is arbitrary in this paper an additional equilibrium relation is developed to govern the behavior of the couples the relation constrained the couple stress tensor to be symmetric and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system on the basis of this modification a linear elastic model for isotropic materials is developed the torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

nodal and multiple solutions of nonlinear problems involving the fractional laplacian abstract this paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional laplacian s u f x u in u 0 on where r n n 2 is a bounded smooth domain s 0 1 s stands for the fractional laplacian when f is superlinear and subcritical we prove the existence of a positive solution a negative solution and a nodal solution if f x u is odd in u we obtain an unbounded sequence of nodal solutions in addition the number of nodal domains of the nodal solutions are investigated

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

large deformation tests and total lagrangian finite element analyses of flexible beams presented here is a total lagrangian displacement based finite element formulation for general anisotropic beams undergoing large displacements and rotations the theory fully accounts for geometric nonlinearities large rotations general initial curvatures and extensionality by using jaumann stress and strain measures an exact coordinate transformation and a new concept of orthogonal virtual rotations moreover transverse shear deformations are accounted for by using a first order shear deformation theory to verify the accuracy of the finite element model two test fixtures have been built for bending and twisting experiments large static deformation tests of beams with different loading conditions have been performed the finite element results agree closely with the experimental results and numerically exact solutions obtained by using a multiple shooting method to solve for post buckling deformations of highly flexible beams undergoing large static rotations and displacements in three dimensional space

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

new conditions on nonlinearity for a periodic schrodinger equation having zero as spectrum abstract we consider the semilinear schrodinger equation u v x u f x u for x r n u x 0 as x where f is a superlinear and subcritical nonlinearity we mainly study the case when both v and f are periodic in x and 0 is a boundary point of a spectral gap of v we extend a linking theorem of kryszewski and szulkin 15 and establish a new variational setting which is more suitable to the above case we obtain two theorems on the existence of ground state solutions with mild assumptions on f

existence and multiplicity of solutions for superlinear fractional schrodinger equations in ℝn in this paper we are concerned with superlinear fractional schrodinger equation su v x u f x u x n where f is continuous when v and f are asymptotically periodic in x we show the existence of ground states when v and f are periodic in x we obtain infinitely many geometrically distinct solutions the method used here is based on the method of nehari manifold and lusternik schnirelmann category theory

bound state for the fractional schrodinger equation with unbounded potential this paper considers the fractional schrodinger equation with unbounded potential we prove the existence of ground state by lagrange multiplier method the standing wave with prescribed frequency is also obtained by nehari s manifold approach in the last part we obtain that the standing wave is bound state

infinitely many solutions of quasilinear schrodinger equation with sign changing potential abstract in this paper we study the following quasilinear schrodinger equation of the form u v x u u 2 u g x u x r n where the potential v x is allowed to be sign changing and the primitive of the nonlinearity g x u is of superlinear growth at infinity in u and is also allowed to be sign changing we obtain the existence of infinitely many nontrivial solutions by using dual approach and mountain pass theorem

on fractional schrodinger equation with periodic and asymptotically periodic conditions abstract in this paper we study the following fractional schrodinger equation s u v x u f x u x r n where s s 0 1 denotes the fractional laplacian under some weaker conditions on the nonlinearity we obtain the existence of solutions for the above problem in periodic case and asymptotically periodic case via variational methods respectively

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

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

shear correction for mindlin type plate and shell elements in this paper two shear correction methods which can guarantee good accuracy suitable for different materials and easy to implement are proposed for the first order theory based plate and shell elements the general variational principles based on the thermodynamic potentials are first derived to provide a uniform frame for the different material non linearity then the two shear correction methods are derived from the obtained general variational principles using the assumed strain and assumed stress methods respectively the shear correction factors scf for homogeneous elastic materials are evaluated using the two methods and the same result of 0 8 is obtained the proposed shear correction method is then applied to an elastic linear hardening plastic deep beam for verification and the conformable result is obtained as compared with the plane strain result copyright 2006 john wiley sons ltd

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation