renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

some existence results for a periodic problem with non smooth potential abstract in this paper two existence results for a class of second order periodic boundary value problems with non smooth potential are obtained we extend the castro lazer thews reduction method to non smooth functionals the obtained result is then exploited to prove the existence of a nontrivial solution furthermore we prove the existence of multiple solutions by using a multiplicity result based on local linking

a class of nonlinear nonhyperbolic systems of conservation laws with well posed initial value problems abstract with mild restrictions on the initial data we show well posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector the obtained weak solutions assume values as borel measures at each time after a smooth solution ceases to exist the concept of entropy density flux functions is extended to such systems finally we show that systems with well posed initial value problems admitting weak solutions of this type necessarily satisfy these structural assumptions

a compression type mountain pass theorem in conical shells in this paper we present a compression type version of the mountain pass lemma in a conical shell with respect to two norms an application to second order ordinary differential equations is included

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

vanishing viscosity with short wave long wave interactions for systems of conservation laws motivated by benney s general theory we propose new models for short wave long wave interactions when the long waves are described by nonlinear systems of conservation laws we prove the strong convergence of the solutions of the vanishing viscosity and short wave long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems we analyze several of the representative examples such as scalar conservation laws general symmetric systems nonlinear elasticity and nonlinear electromagnetism

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

critical points for nondifferentiable functions in presence of splitting a classical critical point theorem in presence of splitting established by brezis nirenberg is extended to functionals which are the sum of a locally lipschitz continuous term and of a convex proper lower semicontinuous function the obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational hemivariational eigenvalue problems

on the riemann problem for a prototype of a mixed type conservation law we study the system of conservation laws given by with initial value the system is elliptic when u2 v2 2 and hyperbolic when u2 v2 2 following liu s construction it is found that the system always has a weak solution which however is not necessarily unique

on a new notion of linking and application to elliptic problems at resonance abstract in this paper we introduce a new notion of linking which includes in particular the notions of homotopical linking and local linking critical point theorems for continuous functionals on metric spaces are presented finally an application to nonlinear elliptic problems at resonance is given

the leray schauder boundary condition in critical point theory abstract we present a version of schechter s bounded mountain pass theorem in a hilbert space not identified to its dual an application to positive solutions of elliptic boundary value problems is included

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

change of type in three phase flow a simple analogue abstract we consider a system of two conservation laws in one space variable which exhibits change of type hyperbolic elliptic the characteristics are complex in a strip in phase space this example has a structure similar to model systems which appear in three phase porous medium flow for example in the saturation equations obtained employing stone s model for the relative permeabilities we solve the riemann problem analytically for initial states in and near the elliptic strip

the palais smale condition versus coercivity possesses a convergent subsequence and cp is said to satisfy ps if ir satisfies ps for every c e ir recently among other results in critical point theory shujie i i showed that if a c functional p x k is bounded from below and satisfies the condition ps then p is coercive shujie s proof uses a gradient flow approach through the so called deformation theorem cf 3 io and for that he needs the notion of a pseudo gradient vector field tjassociated with the functional cp whose existence is guaranteed for c functionals by palais 8 in this paper we present some new results which relate the palais smale condition and the notion of coercivity and are based on the well known variational principle due to ekeland 5 61 in particular a new proof of the above mentioned result of shujie is given it should be pointed out that throughout the paper the given functional p could be assumed to be only gateaux differentiable rather than c and in addition to being conceptually simpler this approach could be used in more general situations where the functional is not even differentiable cf 4 the strong form of ekeland s variational principle to be repeatedly used is given in the following theorem

a class of n nonlinear hyperbolic conservation laws the riemann problem for a class of nonlinear systems of first order hyperbolic conservation laws is studied the class consists of systems where the derivative of the flux function is a lower triangular matrix there are no assumptions on genuine nonlinearity and strict hyperbolicity existence and uniqueness are proved except in a set with measure zero in the phase space and a set with measure zero in the flux function space where there is a one parameter family of solutions travelling waves are used as an entropy condition and examples show that the lax or liu entropy conditions are not sufficient an example shows that the solution does not necessarily depend continuously on the data the model may be used to describe three phase and tracer flow and flow in a neighborhood of a heterogeneity in porous media

multiplicity of nontrivial solutions for quasilinear elliptic equation two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic dirichlet boundary value problem some well known multiplicity results are generalized

homology of saddle point reduction and applications to resonant elliptic systems abstract in the setting of saddle point reduction we prove that the critical groups of the original functional and the reduced functional are isomorphic as application we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity the difficulty that the variational functional does not satisfy the palais smale condition is overcame by taking advantage of saddle point reduction our abstract results on critical groups are crucial

a palais smale approach to lane emden equations we consider the unbounded domain problems u u u p 2 u in u 0i n a ndu 0o n w here is an unbounded domain in r n 2 2 and 2 for n 2 the existence of a ground state solution to the problems is greatly affected by the shape of the domain to determine the existence of the solutions in a general domain remains a challenge task for the flat interior flask domain that consists a strip and a ball attached to the bottom of the strip previous results have asserted the existence of a ground state solution when the diameter of the ball is greater than a positive constant however the existence of the solutions when the diameter of the ball equals to the width of the strip is still an important open question this article resolves the open question partially by considering a variation of the flat interior flask domain which is formed by attaching a stretched ball to the bottom of the strip

existence and multiplicity of periodic solution for non autonomous second order systems with linear nonlinearity abstract the purpose of this paper is to study the existence and multiplicity of periodic solution for the following non autonomous second order systems u t f t u t a e t 0 t u 0 u t u 0 u t 0 some new existence and multiplicity theorems are obtained by using the least action principle and the minimax method

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

nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws we consider the riemann problem for a system of two decoupled nonstrictly hyperbolic burgers like conservation equations with added artificial viscosity we analytically establish two different vanishing viscosity limits for the solution of this system which correspond to the two cases where one of the viscosities vanishes much faster than the other this is done without altering the initial condition as is necessary with travelling wave methods numerical evidence is then provided to show that when the two viscosities vanish at the same rate the solution converges to a limit that lies strictly between the two previously established limits finally we use control theory to explain the mechanism behind this nonuniqueness behavior which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

existence and multiplicity of solutions for a class of semilinear elliptic equations the existence and multiplicity results are obtained for solutions of a class of the dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

a unified proof on existence of homoclinic orbits for some semilinear ordinary differential equations with periodic potentials abstract this paper gives a direct short and unified proof of rabinowitz s theorem grossinho minhos tersian s theorem and tersian chaparova s theorems on existence of homoclinic orbits for second order periodic hamiltonian systems fourth and sixth order periodic ordinary differential equations respectively by brezis nirenberg type mountain pass lemma

existence and multiplicity results for classes of elliptic resonant problems in this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems we also establish several multiplicity results the methods used here are based on combining the minimax methods and the morse theory especially some new observations on the critical groups of a local linking type degenerate critical point

on higher order fully periodic boundary value problems abstract in this paper we present sufficient conditions for the existence of periodic solutions of the higher order fully differential equation u n x f x u x u x u n 1 x with n 3 x a b and f a b r n r a continuous function verifying a nagumo type growth condition a new type of lower and upper solutions eventually non ordered allows us to obtain not only the existence but also some qualitative properties on the solution the last section contains two examples to stress the application to both cases of n odd and n even

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

remarks on the resonant elliptic problem we consider the semilinear elliptic equation u f x u with the dirichlet boundary value for resonant nonlinearity at infinity using the morse theory and perturbation method some existence results of solution and multiple solutions are proved

saddle point techniques from the beginning of the calculus of variations over 300 years ago scientists were interested in finding extrema of expressions now called functionals involving unknown functions it was discovered that functions producing extrema are solutions of differential equations called euler equations in one dimentional problems these were ordinary differential equations and the easiest procedure was to solve the corresponding euler equation and then check if the solution produces the desired maximum or minimum however as researchers began to study higher dimensional problems it became obvious that it was more difficult to solve the euler equations than obtain an extremum of the functional moreover many partial differential equations and systems that were being studied by various scientists happened to be the euler equations of functionals in this case any critical point of the corresponding functional will provide a solution of the given equation this led to the search for critical points of functionals in this paper we describe several methods of finding critical points we present applications

periodic and homoclinic solutions of some semilinear sixth order differential equations in this paper we study the existence of periodic solutions of the sixth order equation uvi auiv bu u u3 0 where the positive constants a and b satisfy the inequality a2 4b the boundary value problem p is considered with the boundary conditions u 0 u 0 uiv 0 0 u l u l uiv l 0 existence of nontrivial solutions for p is proved using a minimization theorem and a multiplicity result using clark s theorem we study also the homoclinic solutions for the sixth order equation uvi auiv bu u a x u u 0 where a is a positive periodic function and is a positive constant the mountain pass theorem of brezis nirenberg and concentration compactness arguments are used

a new critical point theorem for locally lipschitz functionals with applications to differential equations abstract in the present paper a new critical point theorem for locally lipschitz functions is given and as applications one existence and two multiplicity results for zero boundary value problems are obtained several important classical theorems are improved

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

multiple solutions of neumann problem for elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations at resonance or nonresonance with unbounded nonlinearity by the reduction method and the critical point theory in particular a three critical point theorem proposed by brezis nirenberg

multiple periodic solutions for superquadratic second order discrete hamiltonian systems abstract some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second order discrete hamiltonian systems 2 u t 1 f t u t 0 t z by using critical point theory especially a three critical points theorem proposed by brezis and nirenberg

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

periodic solutions of nonautonomous second order hamiltonian systems with even typed potentials abstract some existence and multiplicity theorems are obtained for periodic solutions of nonautonomous second order hamiltonian systems with even typed potentials by the least action principle and the minimax methods in critical point theory

on a notion of category depending on a functional part i theory and application to critical point theory abstract we introduce the notion of category depending on a fixed functional f defined on a topological space this notion permits us to obtain a better lower bound to the number of critical points of f than the bound obtained with the lusternik schnirelman category we also introduce the notion of truncated category depending on the functional f which permits us to obtain a lower bound to the number of critical points of unbounded functionals finally we extend our result after having introduced the notions of relative category and limit relative category depending on the functional f

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

a sphere theorem for manifolds of positive ricci curvature instead of injectivity radius the contractibility radius is estimated for a class of complete manifolds such that ricm 1 km k 2 and the volume of m the volume of the r e ball on the unit m sphere m dim m then for a suitable choice of e e m k every m belonging to this class is homeomorphic to

remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations the existence and multiplicity results are obtained for solutions of a class of the dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

the existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method abstract in this paper we study the existence and multiplicity of classical solutions for a second order impulsive differential equation with periodic boundary conditions by using a variational method and critical point theory we give some new criteria to guarantee that the impulsive problem has at least one solution two solutions and infinitely many solutions when the parameter pair c lies in different intervals respectively some examples are given in this paper to illustrate the main results

an exact riemann solver for compressible two phase flow models containing non conservative products in this article we present a new numerical procedure for solving exactly the riemann problem of compressible two phase flow models containing non conservative products these products appear in the expressions for the interactions between the two phases thus in the compressible limit the governing equations are hyperbolic but can not be written as conservation laws i e in divergence form in general the solution to the riemann problem of these models contains six distinct centered waves according to the relative position of these waves in the x t plane the possible solutions can be classified into four principal configurations the riemann solver we propose herein investigates sequentially each of these configurations until an admissible solution is calculated special configurations corresponding to coalescence of waves are also analyzed and included in the solver further we examine the accuracy and robustness of three known methods for the integration of the non conservative products via a series of numerical tests finally the issue of existence and uniqueness of solutions to the riemann problem is discussed

multiplicity of solutions for resonant neumann problems with an indefinite and unbounded potential we examine semilinear neumann problems driven by the laplacian plus an unbounded and indefinite potential the reaction is a caratheodory function which exhibits linear growth near we allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue and we prove several multiplicity results our approach uses critical point theory morse theory and the reduction method the lyapunov schmidt method