single and multi peak solutions for a nonlinear maxwell schrodinger system with a general nonlinearity abstract for the following elliptic system in r 3 e 2 v v x v x v f v v 2 lim x x 0 we construct positive solutions u and which concentrate at several given isolated local minimum components of v as e 0 the potential v is a strictly positive continuous function and the nonlinearity f is subcritical near infinity and superlinear near zero and satisfies only the berestycki lions condition

semiclassical solutions for choquard equations with berestycki lions type conditions abstract this paper is concerned with existence of multiple semiclassical states for a class of choquard equation under the classic berestycki lions type assumptions assumed on the nonlinearity we obtain multiplicity of semiclassical solutions for the equation by using a method of penalization argument we study further the concentration phenomena of such solutions and show that they converge to the least energy solutions of the associated limit problem as the parameter e goes to 0

multi peak solutions for nonlinear choquard equation in the plane in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

rates of decay to equilibria for p laplacian type equations abstract the long time asymptotics for p laplacian type equations t p m div m p 2 m in r n is studied for p 1 and m n p 1 n p 1 the non negative solutions of the equations are shown to behave asymptotically as t like barenblatt type solutions and the explicit rates of decay are established for the convergence of the relative energy the convergence with respect to the wasserstein distances and the convergence with respect to the l 1 norm the rates are proved to be optimal for p 2 the method used is based on mass transportation inequalities

semi classical ground states concentrating on the nonlinear potential for a dirac equation abstract we study the semi classical limit of the least energy solutions to the nonlinear dirac equation i e k 1 3 k k u a u p x u p 2 u for x r 3 since the dirac operator is unbounded from below and above the associate energy functional is strongly indefinite and since the problem is considered in the global space r 3 the palais smale condition is not satisfied new phenomena and mathematical interests arise in the use of the calculus of variations we prove that the equation has the least energy solutions for all e 0 small and additionally these solutions converge to the least energy solutions of the associate limit problem and concentrate to the maxima of the nonlinear potential p x in certain sense as e 0

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

ground state solutions for a choquard equation with lower critical exponent and local nonlinear perturbation abstract this paper deals with the following choquard equation with a local nonlinear perturbation u u i u n 1 u n 1 u f u x r n u h 1 r n where i r n r is the riesz potential n 3 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and our results improve the previous related ones in the literature

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

phase transitions and symmetry breaking in singular diffusion the long time asymptotics are determined for fast nonlinear diffusion by linearizing otto s gradient descent model at the barenblatt profile the spectrum of the entropy is explicitly determined the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied

concentrating standing waves for the gross pitaevskii equation in trapped dipolar quantum gases abstract we are concerned with the following singularly perturbed gross pitaevskii equation describing bose einstein condensation of trapped dipolar quantum gases e 2 u v x u 1 u 2 u 2 k u 2 u 0 in r 3 u 0 u h 1 r 3 where e is a small positive parameter 1 2 r denotes the convolution k x 1 3 cos 2 x 3 and x is the angle between the dipole axis determined by 0 0 1 and the vector x under certain assumptions on 1 2 r 2 we construct a family of positive solutions u e h 1 r 3 which concentrates around the local minima of v as e 0 our main results extend the results in j byeon and l jeanjean 2007 6 which dealt with singularly perturbed schrodinger equations with a local nonlinearity to the nonlocal gross pitaevskii type equation

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity ii this article continues the analysis of the preceding article showing that the lyapunov functional introduced by newman can be used to prove the stability of the barenblatt pattle solutions of the porous medium equation

ground state and bounded state solution for the nonlinear fractional choquard schrodinger poisson system in this paper we study a class of fractional choquard schrodinger poisson system under some suitable assumptions by using a variational approach we establish the existence of non negative ground state solution and also discuss the nonexistence of ground states moreover we prove the existence of bounded state by using a linking theorem

nonlinear scalar field equations with general nonlinearity abstract consider the nonlinear scalar field equation 0 1 u f u in r n u h 1 r n where n 3 and f satisfies the general berestycki lions conditions we are interested in the existence of positive ground states of nonradial solutions and in the multiplicity of radial and nonradial solutions very recently mederski 2017 made a major advance in that direction through the development in an abstract setting of a new critical point theory for constrained functionals in this paper we propose an alternative more elementary approach which permits to recover mederski s results on 0 1 the keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick and a new decomposition result for bounded palais smale sequences

multi peak solution for nonlinear magnetic choquard type equation in this paper we study a class of nonlinear magnetic choquard type equation involving a magnetic potential and nonlocal nonlinearities by using the method of penalization argument we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential v

existence and asymptotic behavior of vector solutions for coupled nonlinear kirchhoff type systems abstract this paper deals with the following linearly coupled nonlinear kirchhoff type system a 1 b 1 r 3 u 2 d x u 1 u f u v in r 3 a 2 b 2 r 3 v 2 d x v 2 v g v u in r 3 u v h 1 r 3 where a i 0 b i 0 i 0 are constants for i 1 2 0 is a parameter and f g c r r under the general berestycki lions type assumptions on f and g we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods we also study the asymptotic behavior of these solutions as 0

strongly interacting bumps for the schrodinger newton equations we study concentrated bound states of the schrodinger newton equations h2 e x u 0 0 x r3 u 12 2 0 x r3 x 0 u x 0 as x moroz et al an analytical approach to the schrodinger newton equations nonlinearity 12 201 1999 proved the existence and uniqueness of ground states of u 0 0 x r3 u 12 2 0 x r3 x 0 u x 0 as x we first prove that the linearized operator around the unique ground state radial solution 0 u0 with 0 r ae r r 1 o 1 as r x u0 r b r 1 o 1 as r x for some a b 0 has a kernel whose dimension is exactly 3 corresponding to the translational modes using this result we further show that if for some positive integer k the points pi r3 i 1 2 k with pi pj for i j are all local minimum or local maximum or nondegenerate critical points of e p then for h small enough there exist solutions of the schrodinger newton equations with k bumps which concentrate at pi we also prove that given a local maximum point p0 of e p the

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

existence and concentration of positive solutions for a quasilinear equation in rn abstract we study existence and concentration of positive solutions for quasilinear elliptic equations of the form e 2 u e 2 u 2 u v x u h u in r n n 3 as e 0 where the potential v r n r has a positive infimum and inf v inf v for some bounded domain in r n and h is a nonlinearity without some growth conditions such as ambrosetti rabinowitz

multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth abstract in this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in r 2 u v x u 1 x f u x f u x e h x in r 2 where v is a positive continuous potential 0 2 0 2 2 e is a small parameter and f s is the primitive function of f s suppose that the nonlinearity f s is of critical exponential growth in the sense of trudinger moser inequality we prove the existence and multiplicity of solutions by variational methods

concentration phenomena for a class of fractional kirchhoff equations in rn with general nonlinearities abstract in this paper we study the following class of fractional kirchhoff problems e 2 s m e 2 s n u s 2 s u v x u f u in r n u h s r n u 0 in r n where e 0 is a small parameter s 0 1 n 2 s is the fractional laplacian v r n r is a positive continuous function m 0 r is a kirchhoff function satisfying suitable conditions and f r r fulfills berestycki lions type assumptions of subcritical or critical type using suitable variational arguments we prove the existence of a family of positive solutions u e which concentrates at a local minimum of v as e 0

remarks about a fractional choquard equation ground state regularity and polynomial decay abstract with appropriate hypotheses on the nonlinearity f we prove the existence of a ground state solution u for the problem p s u a u p 2 u 1 x f u f u in r n where 0 n p s stands for the s p laplacian operator f is the primitive of f and a is a positive constant when p we also show that u l r n c 0 r n and has polynomial decay

multi peak solutions to kirchhoff equations in r3 with general nonlinearity abstract we are concerned with the existence of multi peak solutions to kirchhoff equation 0 1 2 a b r 3 u 2 d x u v x u f u x r 3 where 0 is a small parameter a b 0 are constants under general conditions of f we construct a family positive solutions u h 1 r 3 which concentrates around the isolated components of positive local minima of v as 0 our result generalize the previous results on single peak solutions to multi peak solutions

on kirchhoff type equations with critical sobolev exponent abstract we study the following brezis nirenberg problem of kirchhoff type a b u 2 d x u u q 2 u u 2 u in u 0 on where r 4 is a bounded domain with the smooth boundary 2 q 4 and a b are positive parameters we obtain some new existence and nonexistence results depending on the values of the above parameters which improve some known results

on nonlocal choquard equations with hardy littlewood sobolev critical exponents abstract we consider the following nonlinear choquard equation with dirichlet boundary condition u u 2 x y d y u 2 2 u f u in where is a smooth bounded domain of r n 0 n 3 0 n and 2 is the critical exponent in the sense of the hardy littlewood sobolev inequality under suitable assumptions on different types of nonlinearities f u we are able to prove some existence and multiplicity results for the equation by variational methods

singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities abstract we consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities the dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect we investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function necessary conditions to locate the possible concentration points are also given

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

concentration for a bi harmonic schrodinger equation with critical nonlinearity abstract this paper concerns the following fourth order problem 4 2 u v x u p x f u u 2 2 u x r n u x 0 as x where 2 2 n n 4 v and p are spatial distributions of external potentials by variational methods the concentration phenomena of the solutions as 0 is considered

existence and concentration of ground state solutions for a critical nonlocal schrodinger equation in r2 we study the following singularly perturbed nonlocal schrodinger equation e2 u v x u e 2 1 x f u f u inr2 where v x is a continuous real function on r2 f s is the primitive of f s 0 2 and e is a positive parameter assuming that the nonlinearity f s has critical exponential growth in the sense of trudinger moser we establish the existence and concentration of solutions by variational methods

existence and multiplicity of solutions for a generalized choquard equation abstract in this paper we study a generalized choquard equation u v x u r n q y f u y x y d y q x f u u h 1 r n where 0 n v and q are linear and nonlinear potentials and f is the primitive function of f when the potentials are periodic and f is odd or even we find infinitely many geometrically distinct solutions using the method of nehari manifold and index theory when the potentials are generalized asymptotically periodic we show the existence of ground states by means of the method of nehari manifold and concentration compactness principle

existence and symmetry of positive ground states for a doubly critical schrodinger system n n 2 and 1 1 satisfying 2 this problem is related to coupled nonlinear schrodinger equations with critical exponent for bose einstein condensate for different ranges of n and 0 we obtain positive ground state solutions via some quite different methods which are all radially symmetric it turns out that the least energy level depends heavily on the relations among and 2 besides for sufficiently small 0 positive solutions are also obtained via a variational perturbation approach note that the palais smale condition can not hold for any positive energy level which makes the study via variational methods rather complicated

bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two abstract we prove using variational methods the existence in dimension two of positive vector ground states solutions for the bose einstein type systems u 1 u 1 u e u 2 1 v e u v 1 in v 2 v 2 v e v 2 1 u e u v 1 in u v h 0 1 where is a bounded smooth domain 1 2 1 the first eigenvalue of h 0 1 1 2 0 and is either positive small or large or negative small the nonlinear interaction between two bose fluids is assumed to be of critical exponential type in the sense of j moser for small solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power like nonlinearities

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

best matching barenblatt profiles are delayed the growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self similar solutions however at next order there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self similar barenblatt solutions this distance can be measured in terms of a relative entropy to the best matching barenblatt profile this best matching barenblatt function determines a scale in new variables based on this scale which are given by a self similar change of variables if and only if the initial datum is one of the barenblatt profiles the typical scale is monotone and has a limit coming back to original variables the best matching barenblatt profile is delayed compared to the self similar solution with same initial second moment as the initial datum such a delay is a new phenomenon which has to be taken into account for instance when fitting experimental data

a note on schrodinger newton systems with decaying electric potential abstract we prove the existence of solutions for the singularly perturbed schrodinger newton system 2 v x u 0 2 u 4 2 0 in r 3 with an electric potential v that decays polynomially fast at infinity the solution concentrates as 0 around structurally stable critical points of the electric potential as a particular case isolated strict extrema of v are allowed

a reduction theorem for time dependent schrodinger operator with finite differentiable unbounded perturbation in this paper we consider the time dependent schrodinger operator with a temporal quasi periodic perturbation which is some function of class cl for any l 100 3n 2 1 by using the kam technique we establish a reduction theorem for the above operator and prove that the invariant tori of the nearly integrable hamiltonian system with unbounded small perturbation are linearly stable both for d d 1 and the limiting case d d 1

multiplicity and concentration of solutions for choquard equations with critical growth abstract in this paper we are concerned with the choquard equation 2 u v x u 3 r 3 u y 6 q y f u y x y d y u 4 u q x f u 6 in r 3 where 0 is a parameter 0 3 6 is the critical exponent in the sense of hardy littlewood sobolev inequality in r 3 v and q are bounded continuous functions f is a continuous subcritical term and f is the primitive function of f by variational methods we establish the existence and concentration of positive ground states and investigate the relation between the number of solutions and the topology of the set where v attains its global minimum and q attains its global maximum

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity i we employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a lyapunov functional which characterizes the evolution of the profile to its self similar solution

the existence and concentration of positive solutions for a nonlinear schrodinger poisson system with critical growth we consider the schrodinger poisson system e2 u v x u x u f u u2 in r3 where the nonlinear term f is of critical growth in this paper we construct a solution u of the above elliptic system which concentrates at an isolated component of positive locally minimum points of v as 0 under certain conditions on f in particular the monotonicity of f s s3 and the so called ambrosetti rabinowitz condition are not required

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

segregated and synchronized vector solutions to linearly coupled systems of schrodinger equations in this paper we study the following linearly coupled system e2 ui pi x ui ui3 j in ijuj ui h1 r3 i 1 n where e 0 is a small parameter pi x are positive potentials and ij ji 0 i j are coupling constants for i j 1 n we investigate the effect of potentials to the structure of the solutions more precisely we construct multi spikes solutions concentrating near the local maximum point x0i of pi x when x0i x0j pi x0i pj x0j a i j i j 1 n the components have spikes clustering at the same point as e 0 when x0i x0j i j the components have spikes clustering at the different points as e 0

critical fractional p kirchhoff type problem with a generalized choquard nonlinearity in this article based on the concentration compactness principle and the mountain pass theorem we establish the existence of positive solutions to the fractional p kirchhoff type problem with a generalized choquard nonlinearity and a critical hardy sobolev term

on concentration of least energy solutions for magnetic critical choquard equations abstract in the present paper we consider the following magnetic nonlinear choquard equation i a x 2 u g x u u x u 2 u 2 2 u in r n u h 1 r n c where n 4 2 2 n n 2 0 n 0 0 is a parameter a x r n r n is a magnetic vector potential and g x is a real valued potential function on r n using variational methods we establish the existence of least energy solution under some suitable conditions moreover the concentration behavior of solutions is also studied as

non stochasticity of time dependent quadratic hamiltonians and the spectra of canonical transformations the authors study the quantum mechanical evolution generated by hamiltonians which are quadratic polynomials in q and p with time dependent coefficients this evolution is determined by a unitary implementation of the phase flow of the corresponding classical hamiltonian in the case of quadratic hamiltonians which are periodic in time the floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum thus the motion is non stochastic in a simple model of a quadratic hamiltonian with random time dependence the quantum mechanical motion is shown to be non stochastic almost surely

existence and concentration of ground states for a choquard equation with competing potentials abstract in this paper we are concerned with the following choquard equation in r 3 that 2 u v x u 3 r 3 p y u y p x y p x u p 2 u r 3 q y u y q x y q x u q 2 u where 0 is a parameter 0 3 6 3 q p 6 the functions v and p are positive and q may be sign changing via variational methods we establish the existence of ground states for small and investigate the concentration behavior of ground states and show that they concentrate at a global minimum point of the least energy function as 0

the logarithmic choquard equation sharp asymptotics and nondegeneracy of the groundstate we derive the asymptotic decay of the unique positive radially symmetric solution to the logarithmic choquard equation u au 12 ln 1 x u 2 uin r2 and we establish its nondegeneracy for the corresponding three dimensional problem the nondegeneracy property of the positive ground state to the choquard equation was proved by e lenzmann 2009 13

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

remarks on a brezis nirenberg s result in this note we revisit the equation with sobolev critical exponent which is firstly studied by brezis nirenberg in 4 p u u u p 2u u 2 2u in u h01 where rn is a bounded smooth domain n 3 2 0 r we show that for any m n there exists m 0 such that p has m solutions for any r 0 and every 0 m when n 4 0 we show that there is 0 0 such that problem p has a nontrivial solution for every 0 moreover we get multiple sign changing solutions for problem p

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

erratum standing waves for nonlinear schrodinger equations with a general nonlinearity erratum arch rational mech anal doi 10 1007 s00205 006 0019 3through answering to some questions from joao marcos bezzerra do oon the paper above we found some mistakes in the paper we thank him forhis interest and careful reading here we correct these mistakes first at the beginning of the proof of proposition 4 14 17 line on page193 the correct version is by compactness of s

ground state of a magnetic nonlinear choquard equation abstract we consider the stationary magnetic nonlinear choquard equation i a x 2 u v x u 1 x f u f u u u where a r n r n is a vector potential v is a scalar potential f r r and f is the primitive of f under mild hypotheses we prove the existence of a ground state solution for this problem we also prove a simple multiplicity result by applying ljusternik schnirelmann methods

singular perturbed kirchhoff type problem with critical exponent abstract in the present paper we consider the following kirchhoff type problem with critical exponent e 2 a e b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 by variational methods we show the existence of the positive solutions concentrating around global minima of the potential v x as e 0 we do not need the monotonicity of the function f 3

uniqueness and nondegeneracy of the ground state for a quasilinear schrodinger equation with a small parameter abstract we study least energy solutions of a quasilinear schrodinger equation with a small parameter we prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory

multiplicity of sign changing solutions for kirchhoff type equations abstract we are concerned with sign changing solutions to nonlocal perturbation of the nonlinear schrodinger equation in the whole r 3 by using the method of invariant sets of descending flow we prove the existence of multiple sign changing solutions provided the perturbation is sufficiently small

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 of multiple semiclassical solutions for a critical choquard equation with indefinite potential abstract we study the existence and multiplicity of semiclassical states for the choquard equation with critical growth e 2 u v x u r n g y u y x y d y g x u in r n where n 3 0 min 4 n v x is sign changing and g is the primitive of g which is of critical growth due to the well known hardy littlewood sobolev inequality under suitable assumptions on v and g we prove the existence and multiplicity of semiclassical states by critical point theory

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

positive solutions for the kirchhoff type problem involving general critical growth part i existence theorem involving general critical growth abstract in this paper we consider the following kirchhoff type problem a r 3 u 2 d x b r 3 u 2 d x u b u f u in r 3 u h 1 r 3 u 0 in r 3 where 0 is a parameter a b are positive constants and f reaches the critical growth without the ambrosetti rabinowitz condition we prove the existence of positive solutions for the kirchhoff type problem with a general critical nonlinearity we also study the asymptotics of solutions as 0 numerical solutions for related problems will be discussed in the second part

existence of semiclassical ground state solutions for a generalized choquard equation abstract in this paper we study a generalized quasilinear choquard equation e p p u v x u p 2 u e n r n q y f u y x y q x f u in r n where p is the p laplacian operator 1 p n v and q are two continuous real functions on r n 0 n f s is the primate function of f s and e is a positive parameter under suitable assumptions on p and f we establish a new concentration behavior of solutions for the quasilinear choquard equation by variational methods the results are also new for the semilinear case p 2

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

semiclassical analysis for pseudo relativistic hartree equations abstract in this paper we study the semiclassical limit for the pseudo relativistic hartree equation e 2 m 2 u v u i u p u p 2 u in r n where m 0 2 p 2 n n 1 v r n r is an external scalar potential i x c n x n is a convolution kernel c n is a positive constant and n 1 p n n for n 3 p 2 our equation becomes the pseudo relativistic hartree equation with coulomb kernel

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

standing waves for nonlinear schrodinger equations with a general nonlinearity for elliptic equations e2 u v x u f u 0 x rn n 3 we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of v as e 0 under conditions on f which we believe to be almost optimal

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

pure point spectrum of the floquet hamiltonian for the quantum harmonic oscillator under time quasi periodic perturbations we prove that the 1 d quantum harmonic oscillator is stable under spatially localized time quasi periodic perturbations on a set of diophantine frequencies of positive measure this proves a conjecture raised by enss veselic in their 1983 paper ev in the general quasi periodic setting the motivation of the present paper also comes from construction of quasi periodic solutions for the corresponding nonlinear equation

long time asymptotics for the porous medium equation the spectrum of the linearized operator we compute the complete spectrum of the displacement hessian operator which is obtained from the confined porous medium equation by linearization around its stationary attractor the barenblatt profile on a formal level the operator is conjugate to the hessian of the entropy via similarity transformation we show that the displacement hessian can be understood as a self adjoint operator and find that its spectrum is purely discrete the knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor more precisely the inspection of the eigenfunctions allows to identify symmetries in rn with flows whose rates of convergence are faster than the uniform translation governed bound the present work complements the analogous study of denzler mccann for the fast diffusion equation

ground state sign changing solutions for a class of subcritical choquard equations with a critical pure power nonlinearity in rn abstract in this paper we investigate the existence of ground state sign changing solutions for a class of choquard equations u 1 f x u i k u p k x u p 2 u u 2 2 u x r n where k and f are nonnegative functions n 3 2 2 n n 2 p 2 n n 2 1 0 and 1 is the first eigenvalue of the equation u u f x u in h 1 r n using the sign changing nehari manifold we prove that the choquard equation has at least one ground state sign changing solution this paper can be regarded as the complementary work of ghimenti and van schaftingen 2016 van schaftingen and xia 2017

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients