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

synthesized affine invariant function for 2d shape recognition by defining the weighted wavelet synthesis the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized affine invariant function saif the synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise contaminated bands the saif is shown excellent in the invariance property and representative in describing the original shape for automated recognition experimental results demonstrate that automated shape recognition based on the saif achieves high correctness and significantly outperforms those using conventional wavelet affine invariant functions

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

asymptotic bep and sep of quadratic diversity combining receivers in correlated ricean fading non gaussian noise and interference in this paper we study the asymptotic behavior of the bit error probability bep and the symbol error probability sep of quadratic diversity combining schemes such as coherent maximum ratio combining mrc differential equal gain combining egc and noncoherent combining nc in correlated ricean fading and non gaussian noise which in our definition also includes interference we provide simple and easy to evaluate asymptotic bep and sep expressions which show that at high signal to noise ratios snrs the performance of the considered combining schemes depends on certain moments of the noise and interference impairing the transmission we derive general rules for calculation of these moments and we provide closed form expressions for the moments of several practically important types of noise such as spatially dependent and spatially independent gaussian mixture noise correlated synchronous and asynchronous co channel interference and correlated gaussian interference from our asymptotic results we conclude that a the asymptotic performance loss of binary frequency shift keying bfsk with nc compared to binary phase shift keying bpsk with mrc is always 6 db independent of the type of noise and the number of diversity branches b the asymptotic performance loss of differential egc compared to mrc is always 3 db for additive white gaussian noise but depends on the number of diversity branches and may be larger or smaller than 3 db for other types of noise and c not only fading correlation but also noise correlation negatively affects the performance of quadratic diversity combiners

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

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

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

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

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

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

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

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 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

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

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

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

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

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

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

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