a new infinite dimensional linking theorem for parameter dependent functionals and periodic schrodinger equations abstract we present a new infinite dimensional linking theorem for parameter dependent functionals this theorem replaces some semi continuous assumptions in the classical infinite dimensional linking theorem with new assumptions and insures the existence of bounded and variant palais smale sequences for a strongly indefinite functional as an application of this theorem we obtain nontrivial solutions for strongly indefinite periodic schrodinger equations with sign changing and asymptotically linear nonlinearities

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

a comparative study of finite element and finite difference methods for cauchy riemann type equations a least squares formulation of the system operatorname div bf u rho operatorname curl bf u zeta is surveyed from the viewpoint of both finite element and finite difference

on hamiltonian elliptic systems with periodic or non periodic potentials abstract this paper is concerned with the following hamiltonian elliptic system u b x u v x u h v x u v v b x v v x v h u x u v for x r n existence and multiplicity of solutions are obtained for the systems with periodic or non periodic potentials v via variational methods

a mixed finite element method for the stokes problem an acceleration pressure formulation the existing finite element methods 6 9 11 for the stokes equation lead to a saddle point problem for removing certain restrictions on the elements and improving the accuracy this paper develops a new method the stokes equation is treated as a first order linear system using a least squares method to minimize the defect in the differential operator we find that the result converges asymptotically to optimal order in sobolev spaces in the 2 d case if the domain is subdivided into regular triangles and the piecewise linear functions in the space h 1 are chosen to be trial functions for both velocity and pressure then we have error estimates p p h 0 u u h 0 ch 2

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

some existence results of solutions for the semilinear elliptic equations on rn abstract in this paper we prove existence of w 1 2 r n solutions for the following equations on r n u q x u u g x u where the operator q x is indefinite and g x z is superlinear or asymptotically linear as z

multiple solutions of schrodinger equations with indefinite linear part and super or asymptotically linear terms abstract based on new information concerning strongly indefinite functionals without palais smale conditions we study existence and multiplicity of solutions of the schrodinger equation u v x u g x u for x r n u x 0 as x where v and g are periodic with respect to x and 0 lies in a gap of v supposing g is asymptotically linear as u and symmetric in u we obtain infinitely many geometrically distinct solutions we also consider the situation where g is super linear with mild assumptions different from those studied previously and establish the existence and multiplicity

some recent results in linking theory and applications abstract the paper collects some recent results in weak linking theorems some applications are given mainly in hamiltonian systems and schrodinger equations a relationship between linking and sign changing critical point is given it is shown that the existence of solutions to elliptic equations with jumping nonlinearities is independent of the fucik spectrum

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

multiple solutions for a class of nonlinear schrodinger equations abstract in this paper we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the schrodinger equation u v x u g x u for x r n assuming v x and g x u depend periodically on x we deal with the situations where g x u is as u asymptotically linear or superlinear with certain hypothesis different from ones used in previous related study our approach is variational and we use the cerami condition instead of the palais smale one for deformation arguments

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

a finite element method for first order elliptic systems in three dimensions abstract we consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions we also discuss div curl type systems in three independent variables both a least squares method and a new subdomain collocation least squares method are considered the analysis yield optimal asymptotic h 1 and l 2 error estimates

homoclinics for asymptotically quadratic and superquadratic hamiltonian systems abstract in this paper we find new conditions which are different from those used in previous related studies to ensure the existence of infinitely many homoclinic orbits for the second order hamiltonian systems of the form q v q t q here we assume that v t q depends periodically on t and assume on q that v t q is asymptotically quadratic at q 0 and is as q either asymptotically quadratic or superquadratic as well as the new conditions

an infinite dimensional linking theorem without upper semi continuous assumption and its applications abstract we obtain a new infinite dimensional linking theorem this theorem replaces the upper semi continuous assumption in the classical infinite dimensional linking theorem of kryszewski and szulkin by a new assumption as an application of this theorem we obtain a nontrivial solution for a strongly indefinite periodic schrodinger equation with sign changing nonlinearity

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

multiple periodic solutions for lattice dynamical systems with superquadratic potentials abstract in this paper we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction the autonomous dynamical system is described by the following infinite system of second order differential equations q i i 1 q i 1 q i i q i q i 1 i z where i denotes the interaction potential between two neighboring particles and q i t is the state of the i th particle supposing i is superquadratic at infinity for all t 0 we obtain a nonzero t periodic solution of finite energy which may be nonconstant in some range of period if in addition i x is even in x we also obtain infinitely many geometrically distinct solutions for any period t 0 in particular a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period since the functional associated to the above system is invariant under the actions of the non compact group z and the continuous compact group s 1 under our assumptions in order to prove our results we need to extend the abstract critical point theorem about strongly indefinite functional developed by bartsch and ding math nachr 279 2006 1267 1288 to a more general class of symmetry

approximation by cp smooth lipschitz functions on banach spaces abstract if x is a banach space with an unconditional basis and admits a c p smooth lipschitz bump function and y is a subset of x then any uniformly continuous function on y can be uniformly approximated by functions which are c p smooth and lipschitz on x

asymptotically linear schrodinger equations with sign changing potential abstract in this paper we show that the following system 0 1 u v x u g x v v v x v f x u x r n possesses at least one non trivial solution pair u v where f x t g x t are continuous functions on r n r and asymptotically linear as t and v x is allowed to be sign changing

a unified analysis of a weighted least squares method for first order systems a unified analysis of a weighted least squares finite element method wlsfem for approximating solutions of a large class of first order differential systems is proposed the method exhibits several advantageous features for example the trial and test functions are not required to satisfy the boundary conditions its discretization results in symmetric and positive definite algebraic systems with condition number o h 2 w 2 and a single piecewise polynomial finite element space may be used for all test and trial functions asymptotic convergence of the least squares approximations with suitable weights is established in a natural norm without requiring extra smoothness of the solutions if instead the solutions are sufficiently regular a priori error estimates can be derived under two suitable assumptions which are related respectively to the symmetric positive systems of friedrichs and first order agmon douglis nirenberg adn elliptic systems numerous model problems fit into these two important systems some selective examples are examined and verified in the unified framework

finite element approximation for grad div type systems in the plane this paper studies a 3 times 3 linear system of differential equations of grad div type with appropriate boundary conditions in constructing numerical approximate solutions to this boundary val

existence of infinitely many solutions for p laplacian equations in rn abstract based on the cohomological linking method for cones and a new direct sum decomposition of w 1 p r n we study the existence and multiplicity of solutions of a p laplacian equation in r n with sign changing potential and subcritical p superlinear nonlinearity our assumptions are mild and different from those studied previously

an abstract approach to multibump solutions of periodic schrodinger equations and applications abstract we review a recent result on nonlinear superposition of zeros of asymptotically additive equivariant vector fields as application we present existence results for multibump solutions of nonlinear periodic schrodinger equations here 0 may be in a gap of the spectrum of the schrodinger operator and the nonlinearity may be of convolution type

periodic solutions for a class of non autonomous hamiltonian systems abstract we consider the existence of nontrivial periodic solutions for a superlinear hamiltonian system h j u a t u h t u 0 u r 2 n t r we prove an abstract result on the existence of a critical point for a real valued functional on a hilbert space via a new deformation theorem different from the works in the literature the new deformation theorem is constructed under the cerami type condition instead of palais smale type condition in addition the main assumption here is weaker than the usual ambrosetti rabinowitz type condition 0 h t u u h t u 2 u r 0 this result extends theorems given by li and willem j math anal appl 189 1995 6 32 and li and szulkin j differential equations 112 1994 226 238

asymptotically periodic superquadratic hamiltonian systems abstract we obtain existence of solutions for the system u v x u f v x u v x r n v v x v f u x u v x r n where n 3 v c r n 0 is periodic and f c 1 r n r 2 r is superquadratic at infinity we consider the case that f is periodic and asymptotically periodic in the proofs we apply variational techniques

morse index and multiple solutions for the asymptotically linear schrodinger type equation abstract we study the nonlinear problem u v x u f x u x r n where the schrodinger type operator v is semibounded and has a finite number of eigenvalues below the infimum of the essential spectrum we classify the linear schrodinger equation prove the semi continuity and monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear schrodinger equations based on an application of the classical ambrosetti rabinowitz symmetric mountain pass theorem

smooth extension of functions on a certain class of non separable banach spaces let us consider a banach space x with the property that every real valued lipschitz function f can be uniformly approximated by a lipschitz c1 smooth function g with lip g clip f with c depending only on the space x this is the case for a banach space x bi lipschitz homeomorphic to a subset of c0 for some set such that the coordinate functions of the homeomorphism are c1 smooth hajek and johanis 2010 then we prove that for every closed subspace y x and every c1 smooth lipschitz function f y r there is a c1 smooth lipschitz respectively extension of f to x we also study c1 smooth extensions of real valued functions defined on closed subsets of x these results extend those given in azagra et al 2010 4 to the class of non separable banach spaces satisfying the above property

quasi least squares finite element method for steady flow and heat transfer with system rotation two quasi least squares finite element schemes based on l2 inner product are proposed to solve a steady navier stokes equations coupled to the energy equation by the boussinesq approximation and augmented by a coriolis forcing term to account for system rotation the resulting nonlinear systems are linearized around a characteristic state resulting in linearized least squares models that yield algebraic systems with symmetric positive definite coefficient matrices existence of solutions are investigated and a priori error estimates are obtained the performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess

dual variational methods and nonvanishing for the nonlinear helmholtz equation we set up a dual variational framework to detect real standing wave solutions of the nonlinear helmholtz equation u k2u q x u p 2u u w2 p rn with n 3 2 n 1 n 1 p 2nn 2 and nonnegative q l rn we prove the existence of nontrivial solutions for periodic q as well as in the case where q x 0 as x in the periodic case a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation the solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

on a class of asymptotically linear elliptic problems in rn under these conditions p is an asymptotically linear problem when this equation is considered in a bounded domain 0 r with say the dirichlet boundary condition there is a large literature on existence and multiplicity results of particular interest is then the case of resonance where s and s is the asymptotic linearization of the problem in other words s d s l 0 l 0 is the operator given by

multiple solutions for a choquard system with periodic potential abstract in this paper we prove the existence of infinitely many geometrically distinct solutions of a choquard system with periodic potential for this purpose we prove a new global critical point theorem without the upper semi continuity condition on the variational functional

existence of positive solutions to quasi linear elliptic equations with exponential growth in the whole euclidean space abstract in this paper a quasi linear elliptic equation in the whole euclidean space is considered the nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of trudinger moser type inequality under some assumptions on the potential and the nonlinearity it is proved that there is a nontrivial positive weak solution to this equation also it is shown that there are two distinct positive weak solutions to a perturbation of the equation the method of proving these results is combining trudinger moser type inequality mountain pass theorem and ekeland s variational principle

on the periodic and asymptotically periodic nonlinear helmholtz equation abstract in the first part of this paper the existence of infinitely many l p standing wave solutions for the nonlinear helmholtz equation u u q x u p 2 u in r n is proven for n 2 and 0 under the assumption that q be a nonnegative periodic and bounded function and the exponent p lies in the helmholtz subcritical range in a second part the existence of a nontrivial solution is shown in the case where the coefficient q is only asymptotically periodic

smoothness in weakly compactly generated banach spaces received august 25 1982 revised april 15 1983 if x is a weakly compactly generated wcg banach space then x admits an equivalent c smooth norm if a wcg banach space x admits a ck smooth function with bounded support then x admits ck smooth partitions of unity

solutions in spectral gaps for a nonlinear equation of schrodinger type abstract in this paper we study the existence of a nontrivial h2 r n solution for an equation of the form u x p x u x f x u x u x x r n r where p l r n is a periodic function we assume that the operator p h2 r n l2 r n l2 r n is strongly indefinite and invertible and that x r r is odd and satisfies some superlinear but subcritical growth conditions we extend the class of nonlinearities which has been studied up to now in particular under standard technical restrictions the existence of a solution is derived when lim x f x s f s 0 exists for all s r if we assume that f x s f s for all s r and a e on r n

analysis of a least squares finite element method for the circular arch problem1this work was supported in part by nsc grant 88 2119 m 214 001 taiwan 1 the stability and convergence of a least squares finite element method for the circular arch problem with shear deformation in a first order system formulation are investigated it is shown that the least squares finite element approximations are stable and convergent in a natural energy norm associated with the least squares functional for the shallow arch case the optimal order of convergence in the h 1 norm for all the unknowns can be achieved uniformly with respect to the small thickness parameter and thus the locking phenomenon does not occur in this case a simple and sharp a posteriori error estimator is also addressed

approximation by lipschitz c p smooth functions on weakly compactly generated banach spaces abstract it is shown that on weakly compactly generated banach spaces which admit a lipschitz c p smooth bump function one can uniformly approximate uniformly continuous bounded real valued functions by lipschitz c p smooth functions this provides a lipschitz version of the classical approximation results of godefroy troyanski whitfield and zizler

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

on a periodic schrodinger equation with nonlocal superlinear part we consider the choquard pekar equation open image in new window and focus on the case of periodic potential v for a large class of even functions w we show existence and multiplicity of solutions essentially the conditions are that 0 is not in the spectrum of the linear part v and that w does not change sign our results carry over to more general nonlinear terms in arbitrary space dimension n 2

existence and bifurcation of solutions for nonlinear perturbations of the periodic schrodinger equation abstract in this paper we apply previously obtained abstract bifurcation results to nonlinear perturbations of the periodic schrodinger equation we show that depending on the sign of the nonlinearity lower or upper end points of the continuous spectrum are bifurcation points the main part of the proof consists of the construction of suitable test functions which are required for the application of the abstract theorem instead of pursuing the notion of an eigenpaket of the differential operator which has successfully been used in the one dimensional case we now exploit a construction based on bloch waves of the linear schrodinger operator

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

smooth approximation of lipschitz functions on riemannian manifolds abstract we show that for every lipschitz function f defined on a separable riemannian manifold m possibly of infinite dimension for every continuous e m 0 and for every positive number r 0 there exists a c smooth lipschitz function g m r such that f p g p e p for every p m and lip g lip f r consequently every separable riemannian manifold is uniformly bumpable we also present some applications of this result such as a general version for separable riemannian manifolds of deville godefroy zizler s smooth variational principle

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

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

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

an asymptotically periodic and asymptotically linear schrodinger equation with indefinite linear part abstract we consider the semilinear schrodinger equation u v x u f x u x r n u h 1 r n where v x is asymptotically periodic and sign changing f x u is asymptotically periodic in x and asymptotically linear as u under some weak assumptions on v and f we prove that the above problem has a nontrivial solution

on the convergence and stability of the standard least squares finite element method for first order elliptic systems a general framework of the theoretical analysis for the convergence and stability of the standard least squares finite element approximations to boundary value problems of first order linear elliptic systems is established in a natural norm with a suitable density assumption the standard least squares method is proved to be convergent without requiring extra smoothness of the exact solutions the method is also shown to be stable with respect to the natural norm some representative problems such as the grad div type problems and the stokes problem are demonstrated

ground state solution for a class of indefinite variational problems with critical growth abstract in this paper we study the existence of ground state solution for an indefinite variational problem of the type p u v x w x u f x u in r n u h 1 r n where n 2 v w r n r and f r n r r are continuous functions verifying some technical conditions and f possesses a critical growth here we will consider the case where the problem is asymptotically periodic that is v is z n periodic w goes to 0 at infinity and f is asymptotically periodic

a note on nonlinear schrodinger equations the relation between spectral gaps and the nonlinearity abstract we establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear schrodinger equations the novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term our goal is to weaken the necessary assumptions to obtain a linking structure to the problem for instance to allow zero being in the spectrum or the nonlinearity being sign changing our main difficulty is to overcome the lack of monotonicity on the nonlinear term as well as the lack of compactness since the domain is unbounded with this purpose we require periodicity on v