concentration and multiplicity of solutions for a fourth order equation with critical nonlinearity abstract in this paper we consider the problem p e 2 u u n 4 n 4 e u u 0 in u u 0 on where is a bounded and smooth domain in r n n 8 and e 0 we analyze the asymptotic behavior of solutions of p e which are minimizing for the sobolev inequality as e 0 and we prove existence of solutions to p e which blow up and concentrate around a critical point of the robin s function finally we show that for e small p e has at least as many solutions as the ljusternik schnirelman category of

a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

generalized eigenvalue problems for p q laplacian with indefinite weight abstract this paper presents existence and non existence results on a positive solution for quasilinear elliptic equations of the form r u r u m r x u r 2 u in with 1 r r and 0 under dirichlet boundary condition where is a bounded domain in r n and m r is a weight function in l admitting sign change we show that existence and non existence of a positive solution depend only on the relation between and the first eigenvalue of r laplacian with weight function m r whence it is independent of the operator r and the parameter 0

multiple positive and 2 nodal symmetric solutions of elliptic problems with critical nonlinearity abstract we consider the problem u a x u f x u 2 2 u in u 0 on where is a bounded smooth domain in r n n 4 2 2 n n 2 is the critical sobolev exponent and a f are continuous functions we assume that a and f are invariant under the action of a group of orthogonal transformations we obtain multiplicity results which contain information about the symmetry and symmetry breaking properties of the solutions and about their nodal domains our results include new multiplicity results for the brezis nirenberg problem u u u 2 2 u in u 0 on

multiple nontrivial solutions for critical growth quasilinear elliptic systems abstract in the present paper we consider the following critical growth quasilinear elliptic system p u i 1 p d u i f u i u i q 2 u i in i 1 k u i 0 on i 1 k by exploiting the nehari manifold and variational methods we obtain the existence and multiplicity results of nontrivial solutions for the system

multiplicity of nodal solutions for a critical quasilinear equation with symmetry abstract we consider the quasilinear problem div u p 2 u u q 2 u u p 2 u 0 in u 0 on where r n is a bounded smooth domain n p 2 0 and p q p np n p we show that if is invariant under a nontrivial orthogonal involution then for sufficiently small there is an effect of the equivariant topology of on the number of solutions which changes sign exactly once

the existence of a nontrivial solution to the p q laplacian problem with nonlinearity asymptotic to up 1 at infinity in rn abstract we consider the following elliptic problem p u m u p 2 u q u n u q 2 u f x u x r n u w 1 p r n w 1 q r n where m n 0 n 3 and 1 q p n f x u u p 1 tends to a positive constant l as u we prove in this paper that the problem possesses a nontrivial solution even if the nonlinearity f x t does not satisfy the ambrosetti rabinowitz condition 0 f x u 0 u f x s d s 1 p f x u u for x u r n r

existence and multiplicity of positive solutions for a class of kirchhoff type problems with singularity abstract in this work we study a class of kirchhoff type problems with singularity and nonlinearity and obtain the uniqueness and existence of positive solutions of those problems by the variational methods furthermore we obtain the multiplicity of positive solutions of those problems by the nehari method

considerations on the hyperbolic complex klein gordon equation this article summarizes and consolidates investigations on hyperbolic complex numbers with respect to the klein gordon equation for fermions and bosons the hyperbolic complex numbers are applied in the sense that complex extensions of groups and algebras are performed not with the complex unit but with the product of complex and hyperbolic unit the modified complexification is the key ingredient for the theory the klein gordon equation is represented in this framework in the form of the first invariant of the poincare group the mass operator in order to emphasize its geometric origin the possibility of new interactions arising from hyperbolic complex gauge transformations is discussed

existence and multiplicity of positive solutions for a class of kirchhoff laplacian type problems this paper deals with the existence and multiplicity of positive solutions for a class of kirchhoff laplacian type problems in whole r3 the multiplicity of solutions is established by variational methods and ljusternik schnirelmann theory

a relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem abstract we consider the quasilinear problem div u p 2 u u p 2 u u q 2 u in u 0 on where r n is a bounded smooth domain 1 p n and p q p np n p we show that if is invariant by a non trivial orthogonal involution then for q close to p the equivariant topology of is related with the number of solutions which change sign exactly once the results complement those of castro and clapp nonlinearity 16 2003 579 590 since we consider subcritical nonlinearities and the quasilinear case without any assumption of symmetry we also extend theorem b in benci and cerami arch rational mech anal 114 1991 79 93 for the quasilinear case and prove that the topology of affects the number of positive solutions

asymptotically dense spherical codes ii laminated spherical codes for pt i see ibid vol 43 no 6 p 1774 85 1997 new spherical codes called laminated spherical codes are constructed in dimensions 2 49 using a technique similar to the construction of laminated lattices each spherical code is recursively constructed from existing spherical codes in one lower dimension laminated spherical codes outperform the best known spherical codes in the minimum distance sense for many code sizes the density of a laminated spherical code approaches the density of the laminated lattice in one lower dimension as the minimum distance approaches zero in particular the three dimensional laminated spherical code is asymptotically optimal in the sense that its density approaches the fejes toth 1959 upper bound as the minimum distance approaches zero laminated spherical codes perform asymptotically as well as wrapped spherical codes in those dimensions where laminated lattices are optimal sphere packings

nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains abstract in this paper we consider the semilinear elliptic problem u u x 2 u k x u 2 2 u in u 0 on where is a smooth bounded domain in r n n 4 0 n 2 2 4 2 2 n n 2 is the critical sobolev exponent k x is a continuous function when and k x are invariant under a group of orthogonal transformations we prove the existence of nodal and positive solutions for 0 1 where 1 is the first dirichlet eigenvalue of x 2 on

multiplicity of solutions for homogeneous elliptic systems with critical growth abstract in this paper we are concerned with the number of nonnegative solutions of the elliptic system u q u u v 1 2 h u u v in v q v u v 1 2 h v u v in u v 0 on where r n is a bounded smooth domain n 4 2 2 n n 2 and q u h u and q v h v are the partial derivatives of the homogeneous functions q h c 1 r 2 r where r 2 0 0 in the proofs we apply variational methods and ljusternik schnirelmann theory

on perturbations of a class of a periodic m laplacian equation with critical growth 1 after the well known results of brezis nirenberg 10 problems involving ellipticequationswithcriticalgrowthinboundeddomainshavebeenstudiedbyseveralauthors see for example 7 15 11 13 17 and the references therein the special features of this class of problems considered in this paper is that it isdened in r

the hyperbolic complexification of quantum groups and the isomorphic relations discusses the quantum group glq n h obtained by a hyperbolic complexification of the quantum group glq n and prove that glq n h is isomorphic to the direct product glq n glq n in particular suq 2 h and suq 1 1 h are isomorphic to slq 2

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

existence of multiple positive solutions for a p q laplacian system with critical nonlinearities abstract in this paper our main purpose is to establish the existence of multiple positive solutions of a p q laplacian system with convex and sobolev critical nonlinearities by some standard variational methods whose key is to construct homotopies between and levels of the functional j and some analytical techniques

on the hyperbolic complex linear symmetry groups and their local gauge transformation actions the hyperbolic complex linear groups and the isomorphic relation between these groups and real linear groups are discussed a local hyperbolic complex gauge symmetry of the hyperbolic complex sesquilinear field is equivalent to some local real gauge symmetry of the real bilinear field

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

on a semi linear schrodinger equation in rn abstract we are concerned with the semi linear schrodinger equation u v x u k x u 2 2 u g x u u h 1 r n where n 3 v x is a continuous function such that the spectrum v x of v x in l 2 r n has a negative part 2 2 n n 2 is the critical sobolev exponent k x is a bounded positive function g is of subcritical growth we prove that under suitable hypotheses the equation has a nontrivial solution

arbitrary many peak solutions for a bi harmonic equation with nearly critical growth abstract let be the unit ball centered at the origin in r n n 6 in this paper we study the following problem 2 u x u p e x u 0 x u 0 u 0 x where p n 4 n 4 0 e 0 we prove that for any k n if e is small enough the above problem has a positive solution u e concentrating at k distinct points tending to the boundary of as e goes to 0

existence and multiplicity of positive solutions for a fourth order elliptic equation we prove existence and multiplicity of solutions for the problem left matrix delta 2u lambda delta u vert u vert 2 2u rm in omega hfill hfill hfill hfill cr u delta u 0 quad rm in omega quad u delta u 0 quad rm on partial omega cr right where omega subset open r n n ges 5 is a bounded regular domain lambda 0 and 2 2n n 4 is the critical sobolev exponent for the embedding of w 2 2 omega into the lebesgue spaces

on a class of singularly perturbed elliptic equations in divergence form existence and multiplicity results the main purpose of this paper is to study the existence of single peaked positive solutions of the singularly perturbed elliptic equation e2div j x u v x u f u inrn where j is a symmetric uniformly elliptic matrix and v is a positive potential possibly unbounded from above if f u up then solutions concentrate at non degenerate critical points of x v x p 1p 1 n2 detj x 1 2

quasilinear asymptotically periodic elliptic equations with critical growth abstract the existence of a nontrivial solution for a class of asymptotically periodic equations involving the p laplacian operator in the euclidean space is established the solutions are obtained by an application of the mountain pass theorem and the concentration compactness principle the lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem

bifurcation from infinity in a nonlinear elliptic equation involving the limiting sobolev exponent we consider an elliptic pde with critical nonlinearity and additonal subcritical and super quadratic nonlinearity with dirichlet boundary conditions on a 3 dimensional smooth and bounded domain we prove the existence of at least p 1 solutions where p is the ljusternik schnirelman category of the domain we prove additional results concerning double peaked solutions on a ringshaped domain in particular we characterize the points at which the solutions blow up as the ringshaped domain becomes thinner or thicker

successive refinement lattice vector quantization lattice vector quantization lvq solves the complexity problem of lbg based vector quantizers yielding very general codebooks however a single stage lvq when applied to high resolution quantization of a vector may result in very large and unwieldy indices making it unsuitable for applications requiring successive refinement the goal of this work is to develop a unified framework for progressive uniform quantization of vectors without having to sacrifice the mean squared error advantage of lattice quantization a successive refinement uniform vector quantization methodology is developed where the codebooks in successive stages are all lattice codebooks each in the shape of the voronoi regions of the lattice at the previous stage such voronoi shaped geometric lattice codebooks are named voronoi lattice vqs vlvq measures of efficiency of successive refinement are developed based on the entropy of the indices transmitted by the vlvqs additionally a constructive method for asymptotically optimal uniform quantization is developed using tree structured subset vlvqs in conjunction with entropy coding the methodology developed here essentially yields the optimal vector counterpart of scalar bitplane wise refinement unfortunately it is not as trivial to implement as in the scalar case furthermore the benefits of asymptotic optimality in tree structured subset vlvqs remain elusive in practical nonasymptotic situations nevertheless because scalar bitplane wise refinement is extensively used in modern wavelet image coders we have applied the vlvq techniques to successively refine vectors of wavelet coefficients in the vector set partitioning vspiht framework the results are compared against spiht and the previous successive approximation wavelet vector quantization sa w vq results of sampson da silva and ghanbari 1996

existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions in this paper the existence and multiplicity results of solutions for the following quasilinear elliptic problem 0 1 div x ap u p 2 u h x u p 2u g x u r 2u x x ap u p 2 u f x u q 2u x are established where is an exterior domain in rn with the compact and smooth boundary by the variational methods we prove that the problem 0 1 has at least two positive solutions at the last part of the paper we also consider the critical case and prove the existence of solution by concentration compactness principle

uniqueness of positive multi lump bound states of nonlinear schrodinger equations in this paper we are concerned with multi lump bound states of the nonlinear schrodinger equation

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

multiplicity of positive solutions for a critical quasilinear elliptic system with concave and convex nonlinearities abstract in this paper by using the lusternik schnirelmann category we obtain a multiplicity result for a quasilinear elliptic system with both concave and convex nonlinearities and critical growth terms in bounded domains

multiple positive solutions for a critical quasilinear elliptic system with concave convex nonlinearities abstract in this paper we consider a quasilinear elliptic system with both concave convex nonlinearities and critical growth terms in bounded domains the existence and multiplicity results of positive solutions are obtained by variational methods

on the local gl 4 r gauge symmetry of hyperbolic complex metrics there exists a construction of hyperbolic complex lorentz frames for any signature we prove gl 4 r to be locally isomorphic to u h by lie algebra theory therefore the local gl 4 r gauge symmetry of hyperbolic complex metric g in the nonsymmetric gravitational theory in fact is a symmetry of rotations of the hyperbolic complex lorentz frames

multiplicity of positive solutions to a p laplacian equation involving critical nonlinearity abstract in this paper we deal with multiplicity of positive solutions to the p laplacian equation of the type div u p 2 u u q 1 u p 1 u w 1 p 0 where r n is a bounded domain n p2 2 p q p np n p and is a positive parameter we prove that there exists 0 such that for each 0 the equation has at least cat positive solutions

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

multiple positive solutions for a critical quasilinear elliptic system abstract the existence of cat positive solutions for the p laplacian system with convex and sobolev critical nonlinearities is obtained by some standard variational methods whose key is to construct homotopies between and levels of the functional j and some analytical techniques

on the number of solutions of nls equations with magnetics fields in expanding domains abstract in this paper we look for multiple weak solutions u c for the complex equation i a x 2 u u f u 2 u in the set r n is a smooth bounded domain 0 is a parameter a is a regular magnetic field and f is a superlinear function with subcritical growth our main result relates for large values of the number of solutions with the topology of the set in the proof we apply minimax methods and ljusternik schnirelmann theory

nodal solutions for singularly perturbed equations with critical exponential growth abstract the existence and concentration behavior of nodal solutions are established for the equation 2 u v z u f u in where is a domain in r 2 not necessarily bounded v is a positive holder continuous function and f c 1 is an odd function having critical exponential growth

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

multiple solutions for an elliptic problem related to vortex pairs abstract let be a bounded domain in r n n 2 a harmonic function in in this paper we study the existence of solutions to the following problem arising in the study of vortex pairs p u u p 1 x u 0 x the set p x u x is called vortex core existence of solutions whose vortex core consists of one component and asymptotic behavior of vortex core were studied by many authors for large recently under the condition that has k strictly local minimum points on the boundary we obtain in this paper that for large enough p has a solution with vortex core consisting of k components by a constructive way

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

on a fourth order elliptic equation involving the critical sobolev exponent the effect of the graph topology abstract in this paper we consider the following nonlinear equation 2 u k u n 4 n 4 e u with u 0 in and u u 0 on where is a smooth bounded domain in r n n 8 k is a smooth positive function in and e is a small positive parameter using the effect of the graph topology of the function k we prove existence multiplicity and concentration of solutions we prove also a nonexistence result

on an elliptic problem with critical nonlinearity in expanding annuli this paper is concerned with an elliptic problem with homogeneous boundary conditions and critical nonlinearity pe u up u 0 onae u 0 on ae whereae x rn e x 1 e are expanding annuli ase 0 n 3 andp 1 2n n 2 is the critical sobolev exponent we compute the difference of topology induced by the critical points at infinity between the level sets of the functional corresponding to pe foresmall enough

a complete classification of bifurcation diagrams for a class of p q laplace equations abstract we study the bifurcation problem of positive solutions for the one dimensional p q laplace equation with nonlinear term u r 1 there are five types of order relations for p q r we investigate the exact shape of the bifurcation curve in each type of the order relation we prove that there are two types of bifurcation curves that are increasing two types that are decreasing and one that is not monotone and turns exactly once moreover we study the asymptotic profile of the normalized solution u x u as u 0 or u where u denotes the l norm of u

existence of multi bump solutions for a class of kirchhoff type problems in r3 using variational methods we establish existence of multi bump solutions for a class of kirchhoff type problems a b r3 u 2dx u v x u f u where f is a continuous function with subcritical growth v x is a critical frequency in the sense that infx r3v x 0 we show that if the zero set of v x has several isolated connected components 1 k such that the interior of i is not empty and i is smooth then for 0 large there exists for any non empty subset j 1 k a bump solution is trapped in a neighborhood of j j j

semiclassical limit for nonlinear schrodinger equations with electromagnetic fields abstract we study the existence of standing waves for a class of nonlinear schrodinger equations in r n with both an electric and a magnetic field under suitable non degeneracy assumptions on the critical points of an auxiliary function related to the electric field we prove the existence and the multiplicity of complex valued solutions in the semiclassical limit we show that in the semiclassical limit the presence of a magnetic field produces a phase in the complex wave but it does not influence the location of peaks of the modulus of these waves

existence of a positive solution for a kirchhoff problem type with critical growth via truncation argument abstract in this paper we shall show an existence result of a positive solution for a kirchhoff problem type in a bounded domain of r n that is for the problem m u 2 d x u f x u u 2 2 u in u 0 on we shall study the asymptotic behavior of this solution when converges to infinity our approach is based on the variational method an appropriated truncated argument and a priori estimates to obtain the solution

an elliptic problem with critical growth in domains with shrinking holes we study an elliptic problem involving critical sobolev exponent in domains with small holes we prove the existence of solutions which blow up like a volcano near the centre of each hole c 2003 elsevier inc all rights reserved

blow up and symmetry of sign changing solutions to some critical elliptic equations abstract in this paper we continue the analysis of the blow up of low energy sign changing solutions of semi linear elliptic equations with critical sobolev exponent started in m ben ayed k el mehdi f pacella blow up and nonexistence of sign changing solutions to the brezis nirenberg problem in dimension three ann inst h poincare anal non lineaire in press in addition we prove axial symmetry results for the same kind of solutions in a ball

positive solutions to classes of infinite semipositone p q laplace problems with nonlinear boundary conditions abstract we consider one dimensional p q laplace problems u h t f u t 0 1 u 0 0 a u 1 g u 1 u 1 where 0 a 0 s s p 2 s s q 2 s 1 p q h c 0 1 0 f c 0 r with lim s 0 f s 0 and g c 0 0 0 such that g r s s is nondecreasing with respect to s 0 classifying the behaviors of f near infinity we establish the existence multiplicity and nonexistence of positive solutions in particular we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s f s s r 1 0 1 r q which is new even in semilinear problems p q 2 the proofs are based on a krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity

multiple positive solutions for a class of kirchhoff type problems involving critical sobolev exponents abstract in this paper we study the multiplicity results of positive solutions for a class of kirchhoff type problems involving critical growth terms with the help of nehari manifold and ljusternik schnirelmann category theory we investigate how the coefficient g x of the critical nonlinearity affects the number of positive solutions of that problem furthermore we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

existence of solution for a degenerate p x laplacian equation in rn abstract using variational methods we establish the existence of nontrivial solutions for the following class of problems div a x u p x 2 u u p x 2 u k x u q x 2 u in r n where p q a k r n r are continuous functions satisfying some conditions and the set x r n a x 0 is not empty

multiplicity of positive solutions for a p q laplacian system with concave and critical nonlinearities abstract in this paper we consider a p q laplacian system with concave and sobolev critical nonlinearities by using the n ehari manifold fibering maps and the lusternik schnirelman category we prove that the problem has at least cat 1 distinct positive solutions where cat denotes the lusternik schnirelman category of in itself

existence of positive solutions for a class of p q elliptic problems with critical growth on rn this paper is concerned with the existence of positive solutions to the class of p q elliptic problems with critical growth type div a u p u p 2 u b u p u p 2u f u u 2u u z 0 x rn where is a positive parameter a r r is a function of c1 class and b f r r are continuous functions

positive solutions to a fourth order elliptic problem by the lusternik schnirelmann category abstract in this paper we consider the fourth order problem 2 u u s 1 u u 2 2 u in u u 0 in u u 0 on where is a smooth bounded domain in r n n 5 and 2 2 n n 4 we assume 2 s 1 2 in case n 8 and 2 2 s 1 2 for the critical dimensions n 5 6 7 then we prove that if has a rich topology described by its lusternik schnirelmann category then the problem has multiple solutions at least as many as cat in case the parameter 0 is sufficiently small

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

extensions of a result by g talenti to p q laplace equations abstract we prove a comparison result for solutions to p q laplace equation via schwarz symmetrization for the p laplace equation the corresponding result is due to giorgio talenti in a special radial case we also prove a reverse comparison result

design performance and complexity analysis of residual trellis coded vector quantizers multistage trellis coded vector quantization ms tcvq is developed as a constrained trellis source coding technique the performance of the two stage tcvq is studied for gaussian sources issues of stage by stage design output alphabet selection and complexity are addressed with emphasis on selecting and partitioning the stage codebooks for a given rate ms tcvq achieves low encoding and storage complexity compared to tcvq and comparisons with same dimensional multistage vector quantization indicate a 0 5 3 db improvement in signal to quantization noise ratio

uniform and piecewise uniform lattice vector quantization for memoryless gaussian and laplacian sources lattice vector quantizer design procedures for nonuniform sources are presented the procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization analytical methods for truncating and scaling lattices to be used in vector quantizations are given and their utility is demonstrated for independent and identically distributed i i d gaussian and laplacian sources an analytical technique for piecewise linear multidimensional compandor designs is evaluated for i i d gaussian and laplacian sources by comparing its performance to that of the other vector quantizers

multiplicity of positive solutions to a p q laplacian equation involving critical nonlinearity abstract in this paper our main purpose is to establish the existence of multiple solutions of a p q laplacian equation with convex and sobolev critical nonlinearity by some standard variational methods whose key is to construct homotopies between and levels of the functional i and some analytical techniques

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

gaussian source coding with spherical codes a fixed rate shape gain quantizer for the memoryless gaussian source is proposed the shape quantizer is constructed from wrapped spherical codes that map a sphere packing in spl ropf sup k 1 onto a sphere in spl ropf sup k and the gain codebook is a globally optimal scalar quantizer a wrapped leech lattice shape quantizer is used to demonstrate a signal to quantization noise ratio within 1 db of the distortion rate function for rates above 1 bit per sample and an improvement over existing techniques of similar complexity an asymptotic analysis of the tradeoff between gain quantization and shape quantization is also given

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

design and analysis of a successively refinable lattice quantizer for i i d gaussian sources a successively refinable quantizer is proposed for independent identically distributed i i d gaussian sources our quantizer is a successively refinable extension of the gaussian source coding algorithm based on wrapped spherical codes designed by hamkins and zeger in ldquogaussian source coding with spherical codes rdquo ieee transactions on information theory vol 4 no 11 pp 2980 2989 november 2002 the proposed successively refinable quantizer is analyzed for the case of high resolution and optimum bit allocation is computed our simulation results show that our proposed quantizer outperforms other successively refinable quantizers designed for gaussian sources

ground state solution for a class of indefinite variational problems with critical growth abstract in this paper we study the existence of ground state solution for an indefinite variational problem of the type p u v x w x u f x u in r n u h 1 r n where n 2 v w r n r and f r n r r are continuous functions verifying some technical conditions and f possesses a critical growth here we will consider the case where the problem is asymptotically periodic that is v is z n periodic w goes to 0 at infinity and f is asymptotically periodic

asymptotically dense spherical codes i wrapped spherical codes a new class of spherical codes called wrapped spherical codes is constructed by wrapping any sphere packing spl lambda in euclidean space onto a finite subset of the unit sphere in one higher dimension the mapping preserves much of the structure of spl lambda and unlike previously proposed maps the density of the wrapped spherical codes approaches the density of spl lambda as the minimum distance approaches zero we show that this implies that the asymptotically maximum spherical coding density is achieved by wrapped spherical codes whenever spl lambda is the densest possible sphere packing

three nontrivial solutions for the p laplacian with a nonsmooth potential abstract we consider a nonlinear elliptic problem driven by the partial p laplacian and with a nonsmooth potential hemivariational inequality using variational techniques based on nonsmooth analysis and degree theoretic arguments for operators of the monotone type we establish the existence of at least three distinct nontrivial smooth solutions

existence of least energy positive negative and nodal solutions for a class of p q problems with potentials vanishing at infinity abstract in this paper we prove the existence of at least three nontrivial solutions for the class of p q problems given by div a u p u p 2 u v x b u p u p 2 u k x f u in r n where n 3 2 p n a b f are c 1 real functions and v k are positive and continuous functions thanks to a hardy type inequality we are able to prove compact embeddings in weighted sobolev spaces and by means of mountain pass theorem we get a positive and a negative ground state weak solution moreover by a minimization argument and a quantitative deformation lemma we find a nodal or sign changing weak solution with two nodal domains or that changes sign exactly once in r n the functions f a and b satisfy suitable growth and monotonicity conditions which allow this result to extend the previous ones to a larger class of p q type problems