existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

existence of ground states for a kirchhoff type problem without 4 superlinear condition abstract we consider the existence of ground state solutions for the kirchhoff type problem a b r n u 2 d x u v x u u p 2 u x r n u h 1 r n where a b 0 n 1 2 3 and 2 p 2 here we are interested in the case that 2 p 4 since the existence of ground state for 4 p 2 is easily obtained by a standard variational argument our method is based on a poho z aev type identity

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

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

existence of solution for a class of nonvariational kirchhoff type problem via dynamical methods abstract in this paper we use dynamical methods to establish the existence of nontrivial solution for the following class of kirchhoff type problem p 1 b u 2 d x u f u h u u x u 0 x where r n n 2 3 is a smooth bounded domain b 0 f r r and h r r n 0 are c 1 functions that satisfy some technical conditions

existence and multiplicity of normalized solutions for the nonlinear kirchhoff type problems abstract in this paper we prove the existence and multiplicity results of solutions with prescribed l 2 norm for a class of kirchhoff type problems a b r 3 u 2 d x u u f u in r 3 where a b 0 are constants r and f c r r to obtain such solutions we look into critical points of the energy functional e b u a 2 r 3 u 2 b 4 r 3 u 2 2 r 3 f u constrained on the l 2 spheres s c u h 1 r 3 u 2 2 c here c 0 and f s 0 s f t d t under some mild assumptions on f we show that critical points of e b unbounded from below on s c exist for c 0 in addition we establish the existence of infinitely many radial critical points u n b of e b on s c provided that f is odd finally the asymptotic behavior of u n b as b 0 is analyzed these conclusions extend some known ones in previous papers

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

virtual element method for a nonlocal elliptic problem of kirchhoff type on polygonal meshes abstract we consider the discretization of the nonlocal elliptic problem of kirchhoff type using the virtual element method vem over polygonal meshes the nonlocal diffusion coefficient is approximated by using the l 2 projection operator which is directly computable from the degrees of freedom however the presence of the nonlocal term reduces the sparsity of the jacobian matrix which would increase the computational burden to avoid this and to retain the sparsity of the jacobian the nonlinear system is replaced by an equivalent system inspired by the work on the fem by gudi 2012 the numerical results show that the proposed methodology yields optimal convergence rate in the l 2 norm theoretical estimates are derived for h 1 norms and are verified by solving a numerical example

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

the existence of least energy nodal solutions for some class of kirchhoff equations and choquard equations in r n abstract in this paper we study the existence of least energy nodal solutions for a class of kirchhoff type problems in r n since the kirchhoff equation is a nonlocal one the variational setting to look for sign changing solutions is different from the local cases by using constrained minimization on the sign changing nehari manifold we prove the kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy as a co product of our approaches we obtain the existence of least energy sign changing solution for choquard equations in r n and show that the sign changing solution has an energy strictly larger than the least energy and less than twice the least energy

perturbed kirchhoff type neumann problems in orlicz sobolev spaces the aim of this paper is to establish the existence of infinitely many solutions for perturbed kirchhoff type non homogeneous neumann problems involving two parameters to be precise we prove that an appropriate oscillating behaviour of the nonlinear term even under small perturbations ensures the existence of infinitely many solutions our approach is based on recent variational methods for smooth functionals defined on orlicz sobolev spaces

semiclassical states for nonlinear schrodinger equations with sign changing potentials we establish the existence and multiplicity of semiclassical bound states of the following nonlinear schrodinger equation e2 u v x u g x u for x rn u x 0as x where v changes sign and g is superlinear with critical or supercritical growth as u

the existence of a ground state solution for a class of kirchhoff type equations in ℝn in this paper we study the following kirchhoff type equation where a b are positive constants and n 1 2 3 under appropriate assumptions on v k and g we obtain a ground state solution by using the approach developed by szulkin and weth in 2010

existence and multiplicity of nontrivial solutions for a class of schrodinger kirchhoff type equations abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u f x u in r n u x 0 as x is studied when n 2 3 4 and v x 1 the existence theorem of nontrivial weak solutions for problem p is obtained when n 1 2 3 and v x is more general two existence theorems of nontrivial weak solutions and a sequence of high energy weak solutions for problem p are obtained

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

beta polytopes and poisson polyhedra f vectors and angles abstract we study random polytopes of the form x 1 x n defined as convex hulls of independent and identically distributed random points x 1 x n in r d with one of the following densities f d x c d 1 x 2 x 1 beta distribution 1 or f d x c d 1 x 2 x r d beta distribution d 2 here c d and c d are normalizing constants this setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases we derive exact and asymptotic formulae for the expected number of k faces of x 1 x n for arbitrary k 0 1 d 1 we prove that for any such k this expected number is strictly monotonically increasing with n also we compute the expected internal and external angles of these polytopes at faces of every dimension and more generally the expected conic intrinsic volumes of their tangent cones by passing to the large n limit in the beta case we compute the expected f vector of the convex hull of poisson point processes with power law intensity function using convex duality we derive exact formulae for the expected number of k faces of the zero cell for a class of isotropic poisson hyperplane tessellations in r d this family includes the zero cell of a classical stationary and isotropic poisson hyperplane tessellation and the typical cell of a stationary poisson voronoi tessellation as special cases in addition we prove precise limit theorems for this f vector in the high dimensional regime as d finally we relate the d dimensional beta and beta distributions to the generalized pareto distributions known in extreme value theory

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

existence and asymptotic behavior of nodal solutions for the kirchhoff type problems in r3 abstract in this paper we study the existence and asymptotic behavior of nodal solutions to the following kirchhoff problem a b r 3 u 2 d x u v x u f x u in r 3 u h 1 r 3 where v x is a smooth function a b are positive constants because the so called nonlocal term r 3 u 2 d x u is involved in the equation the variational functional of the equation has totally different properties from the case of b 0 under suitable construction conditions we prove that for any positive integer k the problem has a sign changing solution u k b which changes signs exactly k times moreover the energy of u k b is strictly increasing in k and for any sequence b n 0 n there is a subsequence b n s such that u k b n s converges in h 1 r 3 to w k as s where w k also changes signs exactly k times and solves the following equation a u v x u f x u in r 3 u h 1 r 3

an approximate algorithm for a kirchhoff wave equation the initial boundary value problem for a hyperbolic type quasilinear equation w tt varphi int 0 pi w x 2dx w xx is considered its solution is derived by means of a numerical algorithm consisting of the projection method and the difference method for approximation with respect to spatial and time variables while the resulting discrete system is solved by an iteration process the accuracy of the proposed algorithm is estimated

dual mixed volumes and the slicing problem we develop a technique using dual mixed volumes to study the isotropic constants of some classes of spaces in particular we recover strengthen and generalize results of ball and junge concerning the isotropic constants of subspaces and quotients of lp and related spaces an extension of these results to negative values of p is also obtained using generalized intersection bodies in particular we show that the isotropic constant of a convex body which is contained in an intersection body is bounded up to a constant by the ratio between the latter s mean radius and the former s volume radius we also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure

ground state solutions for kirchhoff type equations with asymptotically 4 linear nonlinearity this paper is concerned with the following kirchhoff type equation a b v 2 v f x v in v 0 on assuming that the primitive of f is asymptotically 4 linear as v a homeomorphism between the nehari manifold and a subset of unit sphere is constructed based on some observations and new techniques inspired by recent work of szulkin and weth 2010 ground state solutions for the equation above are obtained as well as infinitely many pairs of nontrivial solutions for case f x v is odd with respect to v

multiple solutions for a kirchhoff type problem involving the p x laplacian operator abstract the aim of this paper is to obtain weak solutions for a kirchhoff type problem involving the p x laplacian operator the main tool used for obtaining our result is a recent three critical points theorem established by ricceri

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

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

on the isotropy constant of projections of polytopes the isotropy constant of any d dimensional polytope with n vertices is bounded by cn d where c 0 is a numerical constant

multiple nontrivial solutions to a pp kirchhoff equation abstract in this paper morse theory is used to show the existence of nontrivial solutions to the nonlocal p kirchhoff type quasilinear elliptic equation m u p d x p 1 p u f x u in a smooth bounded domain r n with dirichlet boundary condition

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

multiplicity results for a p laplacian discrete problems of kirchhoff type this paper is concerned with the existence and multiplicity solutions for a discrete boundary value problem of p kirchhoff type our technical approach is based on variational methods

multiple solutions for nonlocal system of p x q x kirchhoff type abstract in this paper we study the existence of solutions for a neumann elliptic system with variable exponents under some suitable conditions and by applying an equivalent variational approach to a recent ricceri s three critical points theorem we established the existence of at least three weak solutions

a sub supersolution approach for a quasilinear kirchhoff equation in this paper we establish an existence result for a quasilinear kirchhoff equation via a sub and supersolution approach by using the minty browder s theorem for pseudomonotone operators theory

on an electromagnetic schrodinger equation with critical growth abstract in this paper we consider the electromagnetic schrodinger problem i a 2 u v x u k x u 2 2 u in h a v 1 r n see the definition in section 1 according as whether the indefinite linear is resonant or nonresonant we structure different direct sums of h a v 1 r n then by applying the linking theorem twice to the corresponding functional we establish the existence results

existence and multiplicity of solutions for fourth order elliptic equations of kirchhoff type with critical growth in ℝn in this paper we deal with the existence and multiplicity of solutions for fourth order elliptic equations of kirchhoff type with critical nonlinearity e 4 2 u e 4 a b r n u 2 d x u v x u u 2 2 u h x u t x n by using lions second concentration compactness principle and concentration compactness principle at infinity to prove that ps condition holds locally and by variational method we prove that it has at least one solution and for any m it has at least m pairs of solutions

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

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

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

intersections of random hypersurfaces and visibility for a wide class of stationary random hypersurfaces in d the notion of the projection body is introduced it turns out that this convex body a very special case of which is matheron s steiner compact associated with a poisson process of hyperplanes contains most of the information concerning certain intersection properties of the random hypersurface while its polar reciprocal set is closely connected with the behaviour of the random hypersurface in visibility problems this enables one to give a unified treatment of several intersection and visibility problems for random hypersurfaces a detailed investigation of the case where the random hypersurface is generated by a poisson process is given separately

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes