constructing solutions for the prescribed scalar curvature problem via local pohozaev identities abstract this paper deals with the following prescribed scalar curvature problem u q y y u n 2 n 2 u 0 y y y r 2 r n 2 where q y is nonnegative and bounded by combining a finite reduction argument and local pohozaev type of identities we prove that if n 5 and q r y has a stable critical point r 0 y 0 with r 0 0 and q r 0 y 0 0 then the above problem has infinitely many solutions whose energy can be made arbitrarily large here instead of estimating directly the derivatives of the reduced functional we apply some local pohozaev identities to locate the concentration points of the bump solutions moreover the concentration points of the bump solutions include a saddle point of q y

shannon entropy at avoided crossings in the quantum transition from order to chaos shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics in order to be able to study jointly this entropy for discrete and continuous probability calculations have been performed on a quantized map the kicked harper map resulting in a different behavior as order chaos transition takes place for the discrete position representation and continuous coherent state representation cases this different behavior is analyzed in terms of the distribution of zeros of the husimi function

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

the webster scalar curvature flow on cr sphere part ii abstract this is the first of two papers in which we prove some properties of the webster scalar curvature flow more precisely we establish the long time existence l p convergence and the blow up analysis for the solution of the flow as a by product we prove the convergence of the cr yamabe flow on the cr sphere the results in this paper will be used to prove a result of prescribing webster scalar curvature on the cr sphere which is the main result of the second paper

a perturbation result for prescribing mean curvature in this article we consider a boundary value problem associated with the conformal deformation of metrics we are interested in the case in which a noncompact group of conformal transformation acts on the equation so that the kazdanwarner condition gives rise to obstruction as in the nirenberg problem the simplest situation is the following let bn 1 be the unit ball in rn 1 with euclidean metric g0 its boundary will be denoted by s n bn 1 let h be a smooth function on s n we study the problem of finding a conformal metric gu u4 n 1 g0 whose scalar curvature vanishes in bn 1 and on the boundary bn 1 the mean curvature is given by h this problem is equivalent to solving the following boundary value problem

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

on the cauchy problem of fourth order nonlinear schrodinger equations abstract in this paper we study the cauchy problem of the fourth order schrodinger equation i t u a u b 2 u u p for dimension 4 where p is an integer greater than 1 with initial data in besov spaces we prove that for any 4 p 2 1 4 n p 4 n q the cauchy problem of this equation is locally well posed in b 2 q s p r n and b 2 q s r n where s p n 2 4 p 1 and s s p and for any 1 q almost global well posedness holds in these spaces if initial data are small we also prove that if a 0 then global well posedness holds in these spaces for small initial data

self focusing with fourth order dispersion we analyze self focusing and singularity formation in the nonlinear schrodinger equation nls with high order dispersion i psi t pm delta q psi psi 2 sigma psi 0 in the isotropic mixed dispersion nls i psi t delta psi epsilon delta 2 psi psi 2 sigma psi 0 and in nonisotropic mixed dispersion nls equations which model propagation in fiber arrays

scalar curvatures on sn soit m n g une variete de riemann compacte de dimension n 3 et r x une fonction lisse sur m n on etudie si r x peut etre la courbure scalaire d une metrique g qui est conforme point par point a la metrique originale g

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

prescribing webster scalar curvature on cr manifolds of negative conformal invariants abstract in this paper we are interested in solving the following partial differential equation u r u f u 1 2 n on a compact strictly pseudo convex cr manifold m of dimension 2 n 1 with n 1 this problem naturally arises when solving the prescribing webster scalar curvature problem on m with the prescribed function f using variational techniques we prove several non existence existence and multiplicity results when the function f is sign changing

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

the concentrated solutions on the nonhomogeneous scalar curvature problem abstract we consider the following scalar curvature equation u 1 e k x u 2 1 in r n if k x is nonhomogeneous at the critical points we give the precise description on the concentrated solutions which contains the necessary and existence on the two peak solutions here we generalize the results of cao noussair yan in 6 to the nonhomogeneous case

moving planes moving spheres and a priori estimates abstract in this paper we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi linear elliptic equations using the method of moving planes we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of lin applying the method of moving spheres we give a different proof for a well known sup inf inequality established by brezis li and shafrir

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

finite dimensional global attractor for a dissipative anisotropic fourth order schrodinger equation abstract we study the long time behavior of the solutions to a nonlinear damped anisotropic fourth order schrodinger type equation in r 2 we prove that this behavior is described by the existence of regular finite dimensional global attractor in the energy space

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

local solvability of elliptic and curvature equations on compact manifolds we discuss the search of local constraints for a nonlinear elliptic operator on a compact manifold prove a new version of fredholm theorem for a class of operators with vanishing index and apply it to the local solbability of some prescribed curvature equations

scalar curvature functions in a conformal class of metrics and conformal transformations on etudie le probleme de prescrire la courbure scalaire dans une classe conforme pour l action du groupe conforme on montre l universalite des conditions d integrabilite dues a j l kazdan et f w warner

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

a perturbation method for spinorial yamabe type equations on s m and its application for m ge 2 we prove the existence of non trivial solutions for a certain kind of nonlinear dirac equations with critical sobolev nonlinearities on s m via a perturbative variational method for the special case m 2 this establishes the existence of a conformal immersion s 2 rightarrow mathbb r 3 with prescribed mean curvature h which is close to a positive constant under an index counting condition on h

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

well posedness of higher order nonlinear schrodinger equations in sobolev spaces hs rn and applications in this paper we establish local well posedness in the sobolev space hs rn with s s0 for a general class of nonlinear dispersive equations of the type tu ip dx u f u where p dx is an elliptic differential operator on rn with a real symbol f u is a nonlinear function which behaves like u u for some constant 0 and s0 is a critical index suggested by a standard scaling argument by using such local result and conservation laws we improve the known and obtain some new global well posedness results for the fourth order nonlinear schrodinger equation i tu a u b 2u c u u

non degeneracy of multi bubbling solutions for the prescribed scalar curvature equations and applications abstract we consider the following prescribed scalar curvature equations in r n 0 1 u k y u 2 1 u 0 in r n u d 1 2 r n where k r is a positive function 2 2 n n 2 we first prove a non degeneracy result for the positive multi bubbling solutions constructed in 26 by using the local pohozaev identities then we use this non degeneracy result to glue together bubbles with different concentration rate to obtain new solutions

a prescribed scalar curvature type equation almost critical manifolds and multiple solutions we present an asymptotic analysis for a perturbed prescribed scalar curvature type equation a major consequence is a non existence result in low dimension conversely we prove an existence result in higher dimensions to this aim we develop a general finite dimensional reduction procedure for perturbed variational functionals the general principle can be useful to discuss some other nonlinear elliptic pde with sobolev critical growth in bounded domains

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

stability of solitons described by nonlinear schrodinger type equations with higher order dispersion abstract steady solitons described by the fourth order nonlinear schrodinger type equations in one two and three dimensions with power law nonlinearities are studied conditions of existence and stability of such solitons are found numerical results demonstrating soliton existence stability and instability are described they agree with the developed theory

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions