multiple equilibria of elastic strings under central forces highly singular nonlinear boundary value problems of the bernoullis in this paper we study the multiplicity of equilibrium states of nonlinearly elastic strings under central forces this problem generalizes that studied by joh bernoulli in 1728 for an inextensible string the theory of elastic strings was developed by jas bernoulli in 1691 1704 cf 111 it is technically more difficult than the related problem of the elastic catenary studied in 2 4 81 because it does not have a convenient set of integrals and because the central force field may well be infinite at its center the complications due to this singularity are magnified by the requirement that the compressive contact force in the string must become infinite where the local ratio of deformed to natural length becomes 0 we find that these problems have a multiplicity of both regular and singular solutions with the analysis of the latter requiring a careful extension of the governing laws of mechanics to handle infinite forces in the rest of this section we formulate the governing equations here we pay special attention to questions of regularity which cannot be handled routinely in virtue of the singularities in the equations for a large class of problems we obtain detailed global information about the qualitative behavior of all regular solutions in section 2 we give a full characterization of singular solutions in section 3 we study purely radial solutions by means

parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear sturm liouville boundary value problems

two global several parameter bifurcation theorems and their applications abstract in this paper we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u l u h u u r m x where m is a positive integer x is a banach space l x x is a positively homogeneous compact operator and h r m x x is compact with h o u near u 0 uniformly on bounded sets we obtain some results involving unilateral global bifurcation two examples of p laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied

symmetry principles for s linear homogeneous and nonhomogeneous equations on presente des principes generaux qui demontre que des equations multiparametres homogenes et non homogenes sont symetriques par rapport a l existence de branches de solutions

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

a priori bounds and existence of positive solutions for semilinear elliptic systems abstract we provide a priori l bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v p and u q for any p q lying on the critical sobolev hyperbola our proof combines moving planes method and rellich pohozaev type identities for systems our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded using these a priori bounds and local and global bifurcation techniques we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system

on the generalized spectrum for second order elliptic systems description qualitative et quantitative detaillee du spectre generalise reel pour les systemes elliptiques du second ordre application de la theorie des perturbations lineaires des methodes de prolongement et de la caracterisation variationnelle des valeurs propres de courant hilbert

global several parameter bifurcation and continuation theorems a unified approach via complementing maps we use the concept of complementing map in order to present a unified approach to the continuation principle global implicit function theorem and global bifurcation results for compact vector fields depending on several parameters

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

complementing maps continuation and global bifurcation we state and indicate some of the consequences of a theorem whose sole assumption is the nonvanishing of the leray schauder degree of a compact vector field and whose conclusions yield multidimensional existence continuation and bifurcation results

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

nonorientability of the index bundle and several parameter bifurcation the universal parameter space for sturm liouville boundary value problems is a two torus we show that the index bundle of the universal family of linear s l boundary value problems is non orientable using this we find a one dimensional set of bifurcation points for nonlinear bvp of sturm liouville type

the intricate global structure of buckled states of compressible columns singularity theory and global multiparameter bifurcation theory are combined to show that nonbifurcating buckled states of compressible nonlinearly elastic columns lie on branches having detailed qualitative properties these properties distinguish the branches globally together with a delicate asymptotic analysis they are used to show that the global disposition of solution branches both bifurcating and not is strikingly different from that for incompressible columns

a generalization of vinogradov s mean value theorem we obtain new upper bounds for the number of integral solutions of a complete system of symmetric equations which may be viewed as a multi dimensional version of the system considered in vinogradov s mean value theorem we then use these bounds to obtain weyl type estimates for an associated exponential sum in several variables finally we apply the hardy littlewood method to obtain asymptotic formulas for the number of solutions of the vinogradov type system and also of a related system connected to the problem of finding linear spaces on hypersurfaces

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 the connectivity properties of the solution set of parametrized families of compact vector fields a useful tool in the study of global continua of solutions of nonlinear partial differential equations is a connectivity result on the fixed points set of a l parameter family of maps that goes back to leray and schauder 171 and it was proved in its full generality by browder b in describing it we shall restrict ourselves to the case of parametrized families of compact vector fields defined in open subsets of banach spaces

a concentration compactness lemma with applications to singular eigenvalue problems let omega subset of or equal to r n be any open set we study the nonlinear eigenvalue problem delta p u lambda v x u p 2 u u epsilon d 0 1 p omega where 1 p n and v is an element of l loc 1 omega may have strong singularities and an indefinite sign the key ingredient is a precised concentration compactness lemma related to v dependent limiting problems this work follows extends and simplifies that of a tertikas 1998 j funct anal 154 42 66 dealing with the positive linear case for omega r n c 1999 academic press

asymptotic estimates for rational linear spaces on hypersurfaces we develop a repeated efficient differencing procedure for estimating mean values of certain multidimensional exponential sums over smooth numbers as a consequence we obtain asymptotic lower bounds for the number of linear spaces of fixed dimension up to a given height lying on the hypersurface defined by an additive equation

global higher bifurcations in coupled systems of nonlinear eigenvalue problems on etablit des conditions pour l existence d une bifurcation n aire globale de dimension elevee dans des systemes generaux de problemes aux valeurs propres non lineaires multiparametre qui conservent la structure de couplage des modeles de lotka volterra diffusifs etat stationnaire

structure and dimension of global branches of solutions to multiparameter nonlinear equations this paper is concerned with the topological dimension of global branches of solutions appearing in different problems of nonlinear analysis in particular multiparameter including infinite dimensional continuation and bifurca tion problems by considering an extension of the notion of essential maps defined on sets and using elementary point set topology we are able to unify and extend in a selfcontained fashion most of the recent results on such problems our theory applies whenever any generalized degree theory with the boundary dependence property may be used but with no need of a gebraic structures our applications to continuation and bifutcation follow from the nontriviality of a local invariant in the stable homotopy group of a sphere and give information on the local dimension and behavior of the sets of solutions of bifurcation points and of continuation points

a homogeneity condition guaranteeing bifurcation in multiparameter nonlinear eigenvalue problems on montre qu un point de bifurcation de multiplicite impaire n est jamais isole dans l ensemble des points de bifurcation on introduit une notion de variete de codimension 1 et on utilise ce concept et une theorie de perturbation pour decrire completement l ensemble des points de bifurcation proches d un point de multiplicite un sous l hypothese que l application a est continue

the asymptotic estimates and hasse principle for multidimensional waring s problem abstract motivated by the asymptotic estimates and hasse principle for multidimensional waring s problem via the circle method we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption the number of variables we need is near optimal by proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields we also establish uniform lower bounds for both singular series and singular integral in f q t we thus obtain asymptotic formulas and the hasse principle for multidimensional waring s problem in f q t via a variant of the circle method

on coupled multiparameter nonlinear elliptic systems on considere le systeme de problemes non lineaires aux valeurs limites de dirichlet lu x f u x v x lv x g u x v x x domaine borne de r n l est un operateur aux derivees partielles lineaire fortement et uniformement elliptique et sont des parametres reels f g r 2 r des applications continues telles que f 0 0 0 g 0 0 on presente une analyse detaillee de l ensemble des solutions du point de vue de la theorie de la bifurcation a plusieurs parametres

existence and asymptotic behavior of ground states for quasilinear singular equations involving hardy sobolev exponents abstract we study the existence and decaying rate of solutions for the quasilinear problem p u x f u x g u in r n u 0 in r n u x 0 where p stands for the p laplacian operator 1 p n r n 0 is continuous and not identically zero 0 is a parameter x is the euclidean norm of x 0 p f g 0 0 are continuous and nondecreasing f has sublinear growth and the hardy sobolev exponent p p n n p bounds the growth of g we deal with variational methods and the lower and upper solutions technique

the influence of elasticity on analysis modern developments on examine plusieurs domaines de l analyse auxquels l elasticite a apporte une contribution importante questions de connectivite de la theorie globale des bifurcations particularites de l analyse non lineaire calcul des variations convergence faible inegalites variationnelles autres contributions

on the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains abstract we study a diffusive logistic equation with nonlinear boundary conditions the equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density specifically the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases the model is motivated by empirical work on the glanville fritillary butterfly we derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support allee effects the analysis leads to eigenvalue problems with nonstandard boundary conditions