a complete classification of bifurcation diagrams of a dirichlet problem with concave convex nonlinearities abstract we study the bifurcation diagrams of positive solutions of the two point boundary value problem u x f u x 0 1 u 1 u 1 0 where f u g u h u g h c 0 c2 0 and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

on maximum typical and generic ranks we show that for several notions of rank including tensor rank waring rank and generalized rank with respect to a projective variety the maximum value of rank is at most twice the generic rank we show that over the real numbers the maximum value of the real rank is at most twice the smallest typical rank which is equal to the complex generic rank

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

orlicz spaces without extreme points abstract orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both the orlicz and the luxemburg norms their subspaces of order continuous elements with the norms induced from the whole orlicz spaces without extreme points in their unit balls are also characterized the well known spaces l 1 and c 0 with unit balls without extreme points are covered by our results moreover a new example of a banach space without extreme points in its unit ball is given see example 1 this is the subspace l 1 l a of order continuous elements of the space l 1 l equipped with the norm x a 0 1 a x t d t whenever 0 a and t 1 a

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

a complete classification of bifurcation diagrams of a p laplacian dirichlet problem we study the bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 1 p y y p 2y p u is the one dimensional p laplacian and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions in addition for p 2 we give interesting examples which show the evolution phenomena of bifurcation diagrams

basic theory of p amemiya norm in orlicz spaces 1 p extreme points and rotundity in orlicz spaces endowed with these norms abstract in any modular space generated by a convex modular we define a family of new norms called p amemiya norms which are equivalent to the orlicz norm as well as to the luxemburg norm next the new case of orlicz spaces is studied carefully the attainable points of the p amemiya norm in orlicz function spaces generated by n functions are discussed the intervals for p amemiya norm attainability are described criteria for extreme points as well as for rotundity of orlicz function spaces endowed with p amemiya norm are given the obtained results unify complete and extend as well the results presented by a number of papers devoted to studying the geometry of orlicz spaces endowed with the luxemburg norm and the orlicz norm separately from our results it follows that there are orlicz spaces which are rotund for p amemiya norm with 1 p only that is they are neither rotund for the luxemburg norm corresponding to the case p nor for orlicz norm corresponding to the case p 1

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

positive solutions for nonhomogeneous elliptic equations involving critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for the dirichlet problem u f z u q 2 u g z u 2 2 u in where 0 1 q 2 2 2 n n 2 0 r n n 3 is a bounded domain with smooth boundary and f g are nonnegative continuous functions on

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

typical rank and indscal dimensionality for symmetric three way arrays of order i 2 2 or i 3 3 a peculiar property of three way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank given their order typical tensorial rank has much been studied over algebraically closed fields however very few results have been found pertaining to the typical rank of three way arrays over the real field these results refer to arrays sampled randomly from continuous distributions arrays that consist of symmetric slices do not fit into this sampling scheme the present paper offers typical rank results over the real field for arrays containing symmetric slices of order 2 2a nd 3 3 symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts this paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of indscal for all cases considered this is indeed true thus a full indscal solution may require fewer dimensions for a symmetric array than a full cp decomposition applied to an asymmetric array of the same size the reverse situation does not seem to arise next we examine in which cases cp solutions inevitably are indscal solutions finally the rank reducing impact of double standardizing the slices is discussed

multiple positive solutions for dirichlet problems involving concave and convex nonlinearities abstract in this paper we study the multiplicity of positive solutions for the following dirichlet problems involving concave and convex nonlinearities u u f x u q 2 u h x u p 2 u in a u h 0 1 a where 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 0 a r is an infinite strip in r n is a bounded domain in r n 1 and f x h x satisfy suitable conditions

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

multiple positive solutions for a class of concave convex elliptic problems in rn involving sign changing weight in this paper we study the multiplicity of positive solutions for the following concave convex elliptic equation u u f x uq 1 g x up 1in rn u 0in rn u h1 rn where 1 q 2 p 2 2 2nn 2 if n 3 2 if n 1 2 and the parameters 0 we assume that f x f x f x is sign changing and g x a x b x where the functions f a and b satisfy suitable conditions

closedness of the set of extreme points of the unit ball in orlicz spaces equipped with the p amemiya norm abstract it is known see the classical book linear operators i by n dunford and j t schwarz 6 that compact nice operators from a banach space x into the space of continuous functions c z r are extreme points of the unit ball of compact operators b k x c z r where z is a compact hausdorff space recall that an operator t is called nice if t z ext b x where the continuous mapping t is defined by t z x t x z for all x x and z z it is evident that if the set ext b x is closed then t is nice if and only if t z 0 ext b x on a dense subset z 0 z in general the set of extreme points ext b x need not to be closed in 1992 a suarez granero and m wisla presented the criteria for closedness of the set of extreme points of the unit ball in the case of orlicz spaces equipped with the luxemburg norm the aim of this paper is to present the criteria for the closedness of the set of extreme points of the unit ball for the wide class of norms the so called p amemiya norms this class includes both the orlicz and the luxemburg norm the orlicz functions that generates orlicz spaces are assumed to be as much general as possible in particular the orlicz functions can vanish outside zero and jump to infinity what follows that the investigated orlicz spaces can contain isomorphic copy of l or be a subspace of l

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

multiple positive solutions for indefinite semilinear elliptic problems involving a critical sobolev exponent in this paper we study the decomposition of the nehari manifold by exploiting the combination of concave and convex nonlinearities the result is subsequently used in conjunction with the ljusternik schnirelmann category and variational methods to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical sobolev exponent

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

a bound for the waring rank of the determinant via syzygies abstract we show that the waring rank of the 3 3 determinant previously known to be between 14 and 18 is at least 15 we use syzygies of the apolar ideal which have not been used in this way before additionally we show that the symmetric cactus rank of the 3 3 permanent is at least 14

multiplicity results for a non local problem with concave and convex nonlinearities abstract we shall prove a multiplicity result for a non local problem with a super critical nonlinearity of the form 1 s u u u p 2 u u q 2 in u 0 on r n where n is a bounded domain 0 s 1 and 1 q 2 p as a consequence of our results for each p 2 there exists 0 such that for each 0 problem 1 has a sequence of solutions with a negative energy this result is new for super critical values of p we shall also explore the existence of symmetric solutions for symmetric domains our methodology is based on a variational principle established by one of the authors that allows one to deal with super critical problems when the standard euler lagrange functional is restricted to certain convex sets

strongly extreme points in orlicz function spaces abstract for any orlicz function and any finite atomless measure a criterion for x from the unit sphere of the orlicz space l equipped with the luxemburg norm to be strongly extreme is given further orlicz spaces l which are isometric to l are characterized

on the rank of a symmetric form abstract we give a lower bound for the degree of a finite apolar subscheme of a complex symmetric form f in terms of the degrees of the generators of the annihilator ideal f in the special case when f is a monomial x 0 d 0 x 1 d 1 x n d n with d 0 d 1 d n 1 d n we deduce that the minimal length of an apolar subscheme of f is d 0 1 d n 1 1 and if d 0 d n then this minimal length coincides with the rank of f

local superlinearity and sublinearity for indefinite semilinear elliptic problems in this paper the usual notions of superlinearity and sublinearity for semilinear problems like du f x u are given a local form and extended to indefinite nonlinearities here f x s is allowed to change sign or to vanish for s near zero as well as for s near infinity some of the well known results of ambrosetti bre zis cerami are partially extended to this context

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

the nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition abstract using the nehari manifold and fibering maps we prove the existence theorem of the nonlinear boundary problem u u u p 2 u for x u n u q 2 u for x on a bounded domain r n we discuss how the nehari manifold changes as changes and show how the existence of solutions depends on the properties of the nehari manifold

multiple positive solutions for the critical kirchhoff type problems involving sign changing weight functions abstract this paper is concerned with the multiplicity of positive solutions for the critical kirchhoff type problems involving indefinite weight functions 0 1 m u 2 d x u q x u q 2 u k x u 2 2 u x u 0 x where is a smooth bounded domain in r n n 3 1 q 2 m s a b s with 0 a 0 and b 0 the weight functions q and k are continuous and changing sign using the nehari manifold fibering maps and ljusternik schnirelmann category we prove that at least two positive solutions for 0 1 exist provided that 1 and 2 4 furthermore by the mountain pass theorem and ekeland s variational principle it is shown that 0 1 possesses at least three positive solutions whenever 2 n 2 including the case that 1 and 2 4 our results generalize some recent results in the literature

multiple positive solutions for a dirichlet problem involving critical sobolev exponent abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the ljusternik schnirelmann category to prove that the existence of multiple positive solutions for a dirichlet problem involving critical sobolev exponent

positive solutions for critical inhomogeneous elliptic problems in non contractible domains in this paper we investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non contractible domains we show that for f epsilon h 1 satisfying a suitable condition and 0 f not equivalent to 0 the dirichlet problem

a comparison of different notions of ranks of symmetric tensors abstract we introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers namely rank border rank catalecticant rank generalized rank scheme length border scheme length extension rank and smoothable rank we analyze the stratification induced by these ranks the mutual relations between these stratifications allow us to describe the hierarchy among all the ranks we show that strict inequalities are possible between rank border rank extension rank and catalecticant rank moreover we show that scheme length generalized rank and extension rank coincide

the solution to the waring problem for monomials and the sum of coprime monomials abstract in this paper we compute the waring rank of any polynomial of the form f i 1 r m i where the m i are pairwise coprime monomials i e gcd m i m j 1 for i j in particular we determine the waring rank of any monomial as an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form as a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials

positive solutions for a kirchhoff type problem involving multiple competitive potentials and critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for a kirchhoff type problem involving several potentials and critical nonlinearities on unbounded domain the first main result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials the second one is that with the help of nehari manifold and ljusternik schnirelmann category the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established moreover we present a sufficient condition for the nonexistence of ground state solution

a complete classification of bifurcation diagrams of classes of multiparameter p laplacian boundary value problems we study the bifurcation diagrams of positive solutions of the multiparameter p laplacian problem p u x f u x 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2y p u is the one dimensional p laplacian f u g u h u and 0 and 0 are two bifurcation parameters 0 and 0 are two given real numbers assuming that functions g and h satisfy hypotheses h1 h3 and h4a resp h1 h3 and h4b for fixed 0 resp 0 we give a classification of totally eight qualitatively different bifurcation diagrams we prove that on the u plane resp u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 resp 0 more precisely we prove the exact multiplicity of positive solutions in addition for all p 1 we give interesting examples which show complete evolution of bifurcation diagrams as resp varies

on the algebraic boundaries among typical ranks for real binary forms abstract we describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight showing that they are dual varieties of suitable coincident root loci

the structure of the solution set of a generalized ambrosetti brezis cerami problem in one space variable abstract we study the structure of solution set of the nonlinear two point boundary value problem u x f u x 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and f u i 1 m a i u q i j 1 n b j u p j satisfies a1 a4 under a1 a4 we prove that there exists 0 such that the problem has exactly two positive solutions for 0 exactly one positive solution for and no positive solution for more precisely we give a complete description of the structure of the solution set

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole