the computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue in this paper we study the existence of a nontrivial solution of the elliptic problem which is a resonant type both at the origin and at infinity with the same higher eigenvalue and the nonlinear term is permitted to be unbounded our main ingredient is the computation of the critical groups

multiplicity results for asymptotically linear elliptic problems at resonance in this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via morse theory and minimax methods some new observations on the critical groups of a local linking type critical point are used to deal with the resonance case at 0

multiple periodic solutions of ordinary differential equations with double resonance abstract in this paper we study by morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin

on superquadratic periodic systems with indefinite linear part abstract we consider a second order periodic system with an indefinite linear part and a potential function which is superquadratic but does not satisfy the ar condition using morse critical groups we show that the system has at least one nontrivial solution

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

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

an elliptic resonance problem with multiple solutions in this paper we study the existence of multiple nontrivial solutions for a semilinear elliptic equation at resonance by the minimax methods and morse theory

positive solutions of superlinear elliptic equations abstract in this paper we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the p s condition or a priori bounds of positive solutions the first solution is proved as the minimal positive solution while the second one is obtained as the limit of a gradient flow whose starting point is properly chosen the dependence of the minimal solution upon a parameter is also considered

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

multiple solutions for the coercive semilinear elliptic equations abstract in this paper we study the semilinear elliptic equations u f x u x u 0 x where r n is a smooth bounded domain by using the minimax methods bifurcation methods conley index theory and morse theory we obtain six nontrivial solutions for the equations with coercive nonlinearities

pairs of nontrivial solutions for resonant neumann problems abstract we study a semilinear neumann problem which is resonant at with respect to any eigenvalue different from the first and the second eigenvalue of n the negative neumann laplacian using a combination of variational methods with morse theoretic techniques we show that the problem has at least two nontrivial smooth solutions

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

partial regularity of strong local minimizers in the multi dimensional calculus of variations let n be a bounded domain and f a given strongly quasiconvex integrand of class c2 satisfying the growth condition f xi le c 1 xi p for some c 0 and 2 p consider the multiple integral i u int omega f nabla u where u w1 p n the main result of the paper is the proof that any strong local minimizer of i is of class c1 loc for any 0 1 on an open set of full n dimensional measure in the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large without such an assumption weak local minimizers need not be partially regular as we show by a class of examples we also briefly discuss the question of existence of strong local minimizers for i and connections of our results to extensions of weierstrass sufficiency theorem to the multi dimensional setting

multiplicity of solutions for a class of the quasilinear elliptic equations at resonance abstract in this paper by morse theory we obtain the existence and multiplicity for a class of the quasilinear elliptic equations at resonance

multiple solutions for semilinear resonance elliptic problems with nonconstant coefficients at infinity in this paper we study a class of semilinear elliptic equations which are completely resonant with nonconstant coefficients at origin and at infinity by applying morse theory and minimax method we obtain the existence of multiple nontrivial solutions for the equations

solutions to a gradient system with resonance at both zero and infinity in this paper we study the existence and multiplicity of nontrivial solutions for a gradient system with resonance at both zero and infinity via morse theory

multiple solutions for semilinear elliptic boundary value problems with double resonance abstract in this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of with zero dirichlet boundary data the methods we use here are morse theory minimax methods and bifurcation theory

nontrivial solutions for resonant difference systems via computations of the critical groups abstract in this paper we study the existence of solutions for a class of resonant difference systems by using the minimax methods and the morse theory we prove that there exists at least one nontrivial solution for the systems

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

regularity results for vectorial minimizers of a class of degenerate convex integrals abstract we establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with p q growth conditions we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity the results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand

morrey regularity results for asymptotically convex variational problems with p q growth abstract we prove some global morrey regularity results for almost minimizers of functionals of the form u f x u d x this regularity is valid up to the boundary provided the boundary data is sufficiently regular the main assumption on f is that for each x the function f x behaves asymptotically like a convex function with p q growth some discontinuous behavior in the first argument is allowed as a main application we establish analogous regularity results for a broad class of systems of nonhomogeneous partial differential equations

partial lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions abstract we consider the partial lipschitz continuity of minimizers of functionals of the form v f x v d v d x where r n is open and bounded the map x u f x u appearing in the above functional is assumed to be holder continuous with respect to its first two arguments of class c 2 in its third argument and asymptotically related to the more regular map x u a x u g as the main novelty is the allowance that even as the map f may retain full dependence on the minimizer u not only its gradient du and the spatial coordinate x

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

existence of a nontrivial solution for a class of hemivariational inequality problems at double resonance abstract in this paper we discuss a class of semilinear elliptic hemivariational inequality problems by using the nonsmooth minimax principle for locally lipschitz functions we establish the existence of a nontrivial solution for the semilinear elliptic hemivariational inequality problem where incomplete double resonance occurs at infinity between two distinct consecutive eigenvalues

multiple solutions for boundary value problems of second order difference equations with resonance in this paper we study the existence of multiple solutions for boundary value problems of second order difference equations with resonance at both infinity and zero by using morse theory critical point theory minimax methods and bifurcation theory

complex wkb evolution of markovian open systems we derive a consistent complex wkb expression for the characteristic function that is the fourier transform of the wigner function of a markovian open system defined by a generic lindblad equation the hamiltonian can be a general function of positions and momenta so the only restriction is that linear coupling to the environment is assumed the propagation is shown to correspond to a classical evolution of the liouville type in a complex double phase space the imaginary part of the double hamiltonian being responsible for decoherence whereas dissipation is included in the real part the theory exact in the quadratic case is designed to describe the decoherent and dissipative evolution of localized states such as the interference terms of a schrodinger cat state as well as the evolution of extended states which were the exclusive subjects of our previous real wkb approximation the present rederivation allows us to interpret the real formalism as a first order classical perturbation of the complex theory and to discuss its validity this is also clarified by a simple cubic hamiltonian that illustrates various levels of approximation derived from the complex wkb theory

pure state correlations chords in phase space the intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations in the case of pure states these are invariant with respect to fourier transformation the overlaps themselves are here studied in terms of the wigner function and its fourier transform i e the characteristic function or chord function unlike the wigner function the chord function need not be real but eventual symmetry with respect to reflections about a phase space point may relate these representations semiclassical approximations for the classical like region of small chords and for large chords are derived these lead to an interpretation of the fourier invariance in terms of conjugate chords the interrelation of large and small sub planck phase space structures previously noted in the literature is thus reinterpreted

nonlinear elliptic equations with a jumping reaction abstract we study a nonlinear nonhomogeneous dirichlet problem driven by the sum of a p laplacian and a laplacian 2 p and a jumping nonlinearity under very general conditions on the reaction and without using the fucik spectrum we show that the problem has at least three nontrivial solutions and we provide sign information for all of them our approach uses critical point theory truncation and comparison techniques and morse theory

features of moyal trajectories we study the moyal evolution of the canonical position and momentum variables we compare it with the classical evolution and show that contrary to what is commonly found in the literature the two dynamics do not coincide we prove that this divergence is quite general by studying hamiltonians of the form p2 2m v q several alternative formulations of moyal dynamics are then suggested we introduce the concept of star function and use it to reformulate the moyal equations in terms of a system of ordinary differential equations on the noncommutative moyal plane we then use this formulation to study the semiclassical expansion of moyal trajectories which is cast in terms of a order by order in recursive hierarchy of i first order partial differential equations as well as ii systems of first order ordinary differential equations the latter formulation is derived independently for analytic hamiltonians as well as for the more general case of locally integrable ones we present various examples i

applications of morse theory to some nonlinear elliptic equations with resonance at zero abstract in this paper we study the existence and multiplicity of solutions for some nonlinear elliptic boundary value problems with resonance at zero by applying morse theory we do not impose additional global condition on the nonlinearities except for a subcritical growth condition

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

symplectic evolution of wigner functions in markovian open systems the wigner function is known to evolve classically under the exclusive action of a quadratic hamiltonian if the system also interacts with the environment through lindblad operators that are complex linear functions of position and momentum then the general evolution is the convolution of a non hamiltonian classical propagation of the wigner function with a phase space gaussian that broadens in time we analyze the consequences of this in the three generic cases of elliptic hyperbolic and parabolic hamiltonians the wigner function always becomes positive in a definite time which does not depend on the initial pure state we observe the influence of classical dynamics and dissipation upon this threshold we also derive an exact formula for the evolving linear entropy as the average of a narrowing gaussian taken over a probability distribution that depends only on the initial state this leads to a long time asymptotic formula for the growth of linear entropy we finally discuss the possibility of recovering the initial state

parity measurements decoherence and spiky wigner functions notwithstanding radical conceptual differences between classical and quantum mechanics it is usually assumed that physical measurements concern observables common to both theories not so with the eigenvalues 1 of the parity operator the effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks the result is a set of definite parity though otherwise mixed here we derive the general form of a parity collapsed state whether pure or mixed the signature of positive or negative parity is a corresponding spike in the wigner function which is sharpened by decoherence we conjecture that states with pure parity always have negative values in their wigner functions

on resonant neumann problems we consider semilinear neumann problems at resonance and prove existence and multiplicity theorems the existence theorems allow resonance with respect to any eigenvalue of the negative neumann laplacian the multiplicity theorems concern problems resonant at 0 the principal eigenvalue or at the first nonzero eigenvalue our approach uses tools from critical point theory and from morse theory

multiple periodic solutions of superlinear ordinary differential equations with a parameter abstract we study the existence of multiple 2 periodic solutions of ordinary differential equation x x f t x with superlinear terms via homological linking and morse theory

besov morrey spaces function space theory and applications to non linear pde this paper is devoted to the analysis of function spaces modeled on besov spaces and their applications to non linear partial differential equations with emphasis on the incompressible isotropic navier stokes system and semilinear heat equations specifically we consider the class introduced by hideo kozono and masao yamazaki of besov spaces based on morrey spaces which we call besov morrey or bm spaces we obtain equivalent representations in terms of the weierstrass semigroup and wavelets and various embeddings in classical spaces we then establish pseudo differential and para differential estimates our results cover non regular and exotic symbols although the heat semigroup is not strongly continuous on morrey spaces we show that its action defines an equivalent norm in particular homogeneous bm spaces belong to a larger class constructed by grzegorz karch to analyze scaling in parabolic equations we compare karch s results with those of kozono and yamazaki and generalize them by obtaining short time existence and uniqueness of solutions for arbitrary data with subcritical regularity we exploit pseudo differential calculus to extend the analysis to compact smooth boundaryless riemannian manifolds bm spaces are defined by means of partitions of unity and coordinate patches and intrinsically in terms of functions of the laplace operator

multiplicity results and qualitative properties for neumann semilinear elliptic problems abstract in this paper we establish the existence of multiple ordered classical solutions for neumann semilinear elliptic problems and provide qualitative information about them no symmetry assumptions are required neither on the nonlinearity nor on the domain for some results the growth of the nonlinearity at infinity is arbitrary we also consider the cases when the nonlinearity is either superlinear or asymptotically linear with strong resonances extensive use is made of the mountain pass theorem and leray schauder topological degree

critical point theorem for asymptotically quadratic functional without compactness in this paper we build an abstract critical point theorem for the asymptotically quadratic functional without compactness application is devoted to the existence of nontrivial periodic solution of second order hamiltonian systems

higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients abstract we study the local regularity of vectorial minimizers of integral functionals with standard p growth we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity under a w 1 n sobolev dependence on the spatial variable of the integrand n being the space dimension we show that the minimizers have the gradient locally in l q for every q p as a consequence they are locally holder continuous for every 1

multiple periodic solutions of the second order hamiltonian systems with superlinear terms abstract we study the existence of 2 periodic solutions of the second order hamiltonian systems x a t x x v x t x with superlinear terms and with saddle structure near the origin some multiplicity results are obtained by using bifurcation method homological linking and morse theory

nontrivial periodic solutions for asymptotically linear resonant difference problem in this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained the methods used here are based on combining the minimax methods and the morse theory especially the observation on the critical groups

existence results for non local operators of elliptic type abstract in this paper we investigate the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions we make use of homological linking and morse theory

partial regularity of solutions to p x laplacian pdes with discontinuous coefficients abstract for r n an open and bounded region we consider solutions u w loc 1 p x r n with n 1 of the p x laplacian system a x d u p x 2 d u 0 a e x where concerning the coefficient function x a x we assume only that a w 1 q l where q 1 is essentially arbitrary this implies that the coefficient in the pde can be highly irregular and yet in spite of this we still recover that u c loc 0 0 for each 0 1 where 0 is a set of full measure due to the variational methodology that we employ our results apply to the more general question of the regularity of the integral functional a x d u p x d x

holder continuity of weak solutions of p laplacian pdes with vmo coefficients abstract we consider solutions u w 1 p r n of the p laplacian pde a x d u p 2 d u 0 for x r n where is open and bounded more generally we consider solutions of the elliptic system a x g a x d u d u d u 0 x as well as minimizers of the functional g a x d u d x in each case the coefficient map a r is only assumed to be of class v m o l which means that it may be discontinuous without assuming that x a x has any weak differentiability we show that u c loc 0 for each 0 1 the preceding results are in fact a corollary of a much more general result which applies to the functional f x u d u d x in case f is only asymptotically convex