multiple positive solutions for semilinear dirichlet problems with sign changing weight function in infinite strip domains abstract in this paper existence and multiplicity results to the following dirichlet problem u u f x u q 1 h x u p 1 in u 0 in u 0 on are established where r r n 1 is bounded smooth domain and n 2 here 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 is a positive real parameter the function f among other conditions can possibly change sign in and the function h satisfies suitable conditions the study is based on the comparison of energy levels on nehari manifold

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

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

grobner deformations connectedness and cohomological dimension abstract in this paper we will compare the connectivity dimension c p i of an ideal i in a polynomial ring p with that of any initial ideal of i generalizing a theorem of kalkbrener and sturmfels m kalkbrener b sturmfels initial complex of prime ideals adv math 116 1995 365 376 we prove that c p lt i min c p i dim p i 1 for each monomial order as a corollary we have that every initial complex of a cohen macaulay ideal is strongly connected our approach is based on the study of the cohomological dimension of an ideal a in a noetherian ring r and its relation with the connectivity dimension of r a in particular we prove a generalized version of a theorem of grothendieck a grothendieck cohomologie locale des faisceaux coherents et theoremes de lefschetz locaux et globaux sga 2 in seminaire de geometrie algebrique du bois marie 1962 as consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine

simplicial ideals 2 linear ideals and arithmetical rank abstract in the first part of this paper we study scrollers and linearly joined varieties scrollers were introduced in barile and morales 2004 bm4 linearly joined varieties are an extension of scrollers and were defined in eisenbud et al 2005 eghp there they proved that scrollers are defined by homogeneous ideals having a 2 linear resolution a particular class of varieties of important interest in classical geometry are cohen macaulay varieties of minimal degree they were classified geometrically by the successive contribution of del pezzo 1885 dp bertini 1907 b and xambo 1981 x and algebraically in barile and morales 2000 bm2 they appear naturally studying the fiber cone of a codimension two toric ideals morales 1995 m gimenez et al 1993 1999 gms1 gms2 barile and morales 1998 bm1 ha 2006 h ha and morales 2009 hm let s be a polynomial ring and i s a homogeneous ideal defining a sequence of linearly joined varieties we compute depth s i we prove that c v depth s i 1 where c v is the connectedness dimension of the algebraic set defined by i we characterize sets of generators of i and give an effective algorithm to find equations as an application we prove that ara i projdim s i in the case where v is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution in the case where v is a union of linear spaces the ideal i can be characterized by a tableau which is an extension of a ferrer or young tableau all these results are new and extend results in barile and morales 2004 bm4 eisenbud et al 2005 eghp

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

multiplicity of positive and nodal solutions for semilinear elliptic equations in infinite strips abstract in this paper we study the multiplicity of positive and nodal solutions of the following semilinear elliptic equation u u g x u p 2 u in a u h 0 1 a where 2 p 2 n n 2 n 3 the parameter 0 a r is an infinite strip in r n and is a bounded domain in r n 1 we assume that g x a x b x is a positive function where the functions a and b are satisfying some suitable conditions

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

quantum planes of weight 1 1 n 1 in this article we study the categories tails a arising from artin schelter regular algebras of global dimension 3 of weight 1 1 n where n is an integer greater than 1 we determine generators and relations for all such algebras and this information is used to classify the points and lines in tails a

monomial and toric ideals associated to ferrers graphs each partition a 1 2 an determines a so called ferrers tableau or equivalently a ferrers bipartite graph its edge ideal dubbed a ferrers ideal is a squarefree monomial ideal that is generated by quadrics we show that such an ideal has a 2 linear minimal free resolution i e it defines a small subscheme in fact we prove that this property characterizes ferrers graphs among bipartite graphs furthermore using a method of bayer and sturmfels we provide an explicit description of the maps in its minimal free resolution this is obtained by associating a suitable polyhedral cell complex to the ideal graph along the way we also determine the irredundant primary decomposition of any ferrers ideal we conclude our analysis by studying several features of toric rings of ferrers graphs in particular we recover establish formulae for the hilbert series the castelnuovo mumford regularity and the multiplicity of these rings while most of the previous works in this highly investigated area of research involve path counting arguments we offer here a new and self contained approach based on results from gorenstein liaison theory

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

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

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

symmetry of positive solutions of semilinear elliptic equations in infinite strip domains abstract in this article we apply the improved moving plane method to prove the symmetry of the solutions of the dirichlet problem u u g u in infinite strip domains with zero boundary condition