long time dynamics for semi relativistic nls and half wave in arbitrary dimension we consider the cauchy problems associated with semi relativistic nls snls and half wave hw in particular we focus on the following two main questions local global cauchy theory existence and stability instability of ground states in between other results we prove the existence and stability of ground states for snls in the l 2 supercritical regime this is in sharp contrast with the instability of ground states for the corresponding hw which is also established along the paper by showing an inflation of norms phenomenon concerning the cauchy theory we show under radial symmetry assumption the following results a local existence result in h 1 for energy subcritical nonlinearity and a global existence result in the l 2 subcritical regime

nondispersive solutions to the l 2 critical half wave equation we consider the focusing l2 critical half wave equation in one space dimension i partial t u d u u 2 u where d denotes the first order fractional derivative standard arguments show that there is a critical threshold m 0 such that all h1 2 solutions with u l 2 0 and the linear momentum p 0 in mathbb r in particular our main result and its proof can be seen as a model scenario of minimal mass blowup for l2 critical nonlinear pdes with nonlocal dispersion

uniqueness and nondegeneracy of positive solutions of delta s u u u p rm in mathbb r n when s is close to 1 we consider the equation delta s u u u p with s in 0 1 in the subcritical range of p we prove that if s is sufficiently close to 1 the equation possesses a unique minimizer which is nondegenerate

spectral properties of the cauchy process on half line and interval we study the spectral properties of the transition semigroup of the killed one dimensional cauchy process on the half line 0 a nd the interval 1 1 this process is related to the square root of one dimensional laplacian a d 2 dx 2 with a dirichlet exterior condition on a complement of a domain and to a mixed steklov problem in the half plane for the half line an explicit formula for generalized eigenfunctions of a is derived and then used to construct a spectral representation of a explicit formulas for the transition density of the killed cauchy process on the half line or the heat kernel of a in 0 and for the distribution of the first exit time from the half line follow the formula for is also used to construct approximations to eigenfunctions of a in the interval for the eigenvalues n of a in the interval the asymptotic formula n n 2 8 o 1 n is derived and all eigenvalues n are proved to be simple finally efficient numerical methods of estimation of eigenvalues n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

blowup for fractional nls abstract we consider fractional nls with focusing power type nonlinearity i t u s u u 2 u t x r r n where 1 2 s 1 and 0 for s n 2 and 0 2 s n 2 s for s n 2 we prove a general criterion for blowup of radial solutions in r n with n 2 for l 2 supercritical and l 2 critical powers 2 s n in addition we study the case of fractional nls posed on a bounded star shaped domain r n in any dimension n 1 and subject to exterior dirichlet conditions in this setting we prove a general blowup result without imposing any symmetry assumption on u t x for the blowup proof in r n we derive a localized virial estimate for fractional nls in r n which uses balakrishnan s formula for the fractional laplacian s from semigroup theory in the setting of bounded domains we use a pohozaev type estimate for the fractional laplacian to prove blowup

radial symmetry of solution for fractional p laplacian system abstract in this paper we investigate the method of moving planes for fractional p laplacian system we firstly discuss the key ingredients for the method of moving planes such as maximum principle for anti symmetric functions decay at infinity and boundary estimate then we apply the method of moving planes to establish the radial symmetry and the monotonicity of the positive solutions for fractional p laplacian system in a unit ball or in the whole space

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

uniqueness of positive radial solutions of δu f u 0 in ℝ n ii we prove a uniqueness result for the positive solution of u f u 0 in r n which goes to 0 at the result applies to a wide class of nonlinear functions f including the important model case f u u u p 1 p n 2 n 2 the result is proved by reducing to an initial boundary problem for the ode u n 1 r f u 0 and using a shooting method

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

eigenvalues of the fractional laplace operator in the interval abstract two term weyl type asymptotic law for the eigenvalues of the one dimensional fractional laplace operator 2 0 2 in the interval 1 1 is given the n th eigenvalue is equal to n 2 2 8 o 1 n simplicity of eigenvalues is proved for 1 2 l 2 and l properties of eigenfunctions are studied we also give precise numerical bounds for the first few eigenvalues

on the uniqueness of solutions of a nonlocal elliptic system we consider the following elliptic system with fractional laplacian begin aligned delta su uv 2 delta sv vu 2 u v 0 mathrm on mathbb r n end aligned where s in 0 1 and delta s is the s lapalcian we first prove that all positive solutions must have polynomial bound then we use the almgren monotonicity formula to perform a blown down analysis finally we use the method of moving planes to prove the uniqueness of the one dimensional profile up to translation and scaling

existence and mass concentration of pseudo relativistic hartree equation in this paper we investigate the constrained minimization problem e a inf u h u 22 1 ea u where the energy functional ea u r3 u m2 u vu2 dx a2 r3 x 1 u2 u2 dx with m r a 0 is defined on a sobolev space h we show that there exists a threshold a 0 so that e a is achieved if 0 a a and has no minimizers if a a we also investigate the asymptotic behavior of non negative minimizers of e a as a a

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

integral representation of solutions using green function for fractional hardy equations abstract our main aim is to study green function for the fractional hardy operator p s x 2 s in r n where 0 n s and n s is the best constant in the fractional hardy inequality using green function we also show that the integral representation of the weak solution holds

symmetry of components and liouville type theorems for semilinear elliptic systems involving the fractional laplacian abstract in this paper we study the symmetry of components of positive solutions for the fractional elliptic systems in the whole space r n as an application we prove the uniqueness of positive solutions for fractional schrodinger type systems with critical exponents we also prove the symmetry of components and some liouville type theorems for a class of fractional noncooperative systems which contains the systems arising in bose einstein condensation and in biological models of lotka volterra type

stability of solutions for nonlocal problems abstract in this paper we deal with the stability of solutions to fractional p laplace problems with nonlinear sources when the fractional parameter s goes to 1 we prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of p fractional problems in bounded and unbounded domains as s goes to 1 moreover our result applies to treat the stability of p fractional eigenvalues as s goes to 1

the cauchy process and the steklov problem let xt be a cauchy process in rd d 1 we investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain d we establish a connection between the semigroup of this process and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem using this we derive a variational characterization for the eigenvalues of the cauchy process in d this characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the cauchy process inspired by those for brownian motion our results are new even in the simplest geometric setting of the interval 1 1 where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions such results although trivial for the laplacian take considerable work to prove for the cauchy processes and remain open for general symmetric stable processes along the way we present other general properties of the eigenfunctions such as real analyticity which even though well known in the case of the laplacian are not available for more general symmetric stable processes

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

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

on the structure of the nodal set and asymptotics of least energy sign changing radial solutions of the fractional brezis nirenberg problem abstract in this paper we study the asymptotic and qualitative properties of least energy radial sign changing solutions of the fractional brezis nirenberg problem ruled by the s laplacian in a ball of r n when s 0 1 and n 6 s as usual is the positive parameter in the linear part in u we prove that for sufficiently small such solutions cannot vanish at the origin we show that they change sign at most twice and their zeros coincide with the sign changes moreover when s is close to 1 such solutions change sign exactly once finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a tower of bubbles as 0 i e the positive and negative parts concentrate at the same point with different concentration speeds

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

finite time blow up for the fractional critical heat equation in rn abstract we consider the fractional critical heat equation u t s u u n 2 s n 2 s in r n 0 t u x 0 u 0 x in r n where 4 s n 6 s 0 s 1 for sufficiently small t 0 and given k distinct points q 1 q k we show that there exists an initial condition u 0 such that the solution u x t blows up at these k distinct points as t t more precisely the blow up profile around each concentration point takes the form of sharply scaled bubble and u t l r n t t n 2 s 6 s n as t t

the fractional schrodinger equation with hardy type potentials and sign changing nonlinearities abstract we look for solutions to a fractional schrodinger equation of the following form 2 u v x x u f x u k x u q 2 u o n r n 0 where v is bounded and close to periodic potential and x is a hardy type potential we assume that v is positive and f has the subcritical growth but not higher than u q 2 u if is positive and small enough we find a ground state solution i e a critical point of the energy being minimizer on the nehari manifold if is negative we show that there is no ground state solutions we are also interested in an asymptotic behaviour of solutions as 0 and k 0

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

on a fractional magnetic schrodinger equation in r with exponential critical growth abstract we deal with a class of fractional magnetic schrodinger equations in the whole line with exponential critical growth under a local condition on the potential we use penalization methods and ljusternik schnirelmann category theory to investigate the existence multiplicity and concentration of nontrivial weak solutions

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

conflict models in graph form solution concepts and their interrelationships abstract a comprehensive approach for modeling conflicts is the recently developed graph form of conflict models within the flexible structure of the graph model solution concepts including nash stability the various metagame techniques the sequential stability method of fraser and hipel and stackelberg equilibrium are defined and characterized these stability principles are compared with the limited move and non myopic stabilities developed previously for the graph model framework and mathematical interrelationships in this very general context are derived as well some steps are taken toward the development of a taxonomy of solution concepts for the graph form

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

bound state for fractional schrodinger equation with saturable nonlinearity in this paper we study the existence of bound state for the following fractional schrodinger equation p u v x u f u x r n n 3 where with 0 1 is the fractional laplace operator defined as a pseudo differential operator with the symbol 2 v x is a positive potential function and the nonlinearity f is saturable that is f u u l 0 as u by using a variant version of mountain pass theorem we prove that there exists a bound state and ground state of p when v and f satisfy suitable assumptions

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case