nontrivial solution for a semilinear elliptic equation in unbounded domain with critical sobolev exponent where is an unbounded domains with smooth boundary in r 2 2n n 2 a x c1 satisfies the following conditions a1 a ln 2 a2 x a x 0 such that b 4 when is a bounded domain with smooth boundary in r n 5 similar problem has been studied by many mathematicians for example in 3 brezis and nirenberg studied problem p in 4 capozzi et al prove that when

local rigidity of quasi regular varieties for a g variety x with an open orbit we define its boundary x as the complement of the open orbit the action sheaf s x is the subsheaf of the tangent sheaf made of vector fields tangent to x we prove for a large family of smooth spherical varieties the vanishing of the cohomology groups h i x s x for i 0 extending results of bien and brion compos math 104 1 26 1996 we apply these results to study the local rigidity of the smooth projective varieties with picard number one classified in pasquier math ann in press

on fano complete intersections in rational homogeneous varieties complete intersections inside rational homogeneous varieties provide interesting examples of fano manifolds for example if x cap i 1 r d i subset g p is a smooth complete intersection of r ample divisors such that k g p otimes mathcal o g p sum i d i is ample then x is fano we first classify these fano complete intersections which are locally rigid it turns out that most of them are hyperplane sections we then classify general hyperplane sections which are quasi homogeneous

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

quasilinear elliptic equations on noncompact riemannian manifolds abstract the existence of solutions to a class of quasilinear elliptic problems on noncompact riemannian manifolds with finite volume is investigated boundary value problems with homogeneous neumann conditions in possibly irregular euclidean domains are included as a special instance a nontrivial solution is shown to exist under an unconventional growth condition on the right hand side which depends on the geometry of the underlying manifold the identification of the critical growth is a crucial step in our analysis and entails the use of the isocapacitary function of the manifold a condition involving its isoperimetric function is also provided

on some smooth projective two orbit varieties with picard number 1 we classify all smooth projective horospherical varieties with picard number 1 we prove that the automorphism group of any such variety x acts with at most two orbits and that this group still acts with only two orbits on x blown up at the closed orbit we characterize all smooth projective two orbit varieties with picard number 1 that satisfy this latter property

nontrivial solutions for neumann problems with an indefinite linear part abstract in this paper we consider a semilinear neumann problem with an indefinite linear part and a caratheodory nonlinearity which is superlinear near infinity and near zero but does not satisfy the ambrosetti rabinowitz condition using an abstract existence theorem for c 1 functions having a local linking at the origin we establish the existence of at least one nontrivial smooth solution

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

solutions to semilinear elliptic problems with combined nonlinearities abstract we are concerned with the following nonlinear dirichlet problem u h x u q f x u 0 u h 0 1 0 where is a bounded smooth domain in rn n 1 and h x l f x s is asymptotically linear with respect to s at infinity by a variant version of mountain pass theorem we prove that there exist at least two nonnegative solutions under suitable assumptions on f x s our methods also work for the cases where f x s is superlinear or linear in s

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

existence of solutions for an elliptic equation with indefinite weight abstract we consider the elliptic equation with indefinite weight u v x u f x u u h 1 r n where v x is a function possibly changing sign in r n under certain assumptions on f x u we obtain the existence of m n pairs of nontrivial solutions where m n and the m and n are all integers

pseudo effective and nef cones on spherical varieties we show that nef cycle classes on smooth complete spherical varieties are effective and the products of nef cycle classes are also nef let x be a smooth projective spherical variety such that its effective cycle classes of codimension k are nef where 1 le k le text dim x 1 we study the properties of x in particular if x is a toric variety then x is isomorphic to the product of some projective spaces if x is toroidal then x is isomorphic to a rational homogeneous space if x is horospherical text dim x ge 3 and k 2 then effective divisors on x are nef if x is horospherical and effective divisors on x are nef then there is a morphism from x to a rational homogeneous space such that each fiber is isomorphic to the product of some horospherical varieties of picard number one

the cones of effective cycles on projective bundles over curves the study of the cones of curves or divisors on complete varieties is a classical subject in algebraic geometry cf 4 9 10 and it still is an active research topic cf 1 11 or 2 however little is known if we pass to higher co dimension in this paper we study this problem in the case of projective bundles over curves and describe the cones of effective cycles in terms of the numerical data appearing in a harder narasimhan filtration this generalizes to higher codimension results of miyaoka and others 3 15 for the case of divisors an application to projective bundles over a smooth base of arbitrary dimension is also given given a smooth complex projective variety x of dimension n consider the vector spaces

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

a topological method for vector valued and nth order nonlinear boundary value problems has recently attracted the interest of many authors e g 1 4 5 8 111 for the case in which n 1 the prevalent approaches have been the topological method of wazewski 1 81 the shooting method via the maximum principle and the kneser hukuhara continuum theorem 11 a common ingredient in these approaches is the use of upper and lower solutions to obtain bounds on the solutions recently jachson and palamides 5 and palamides lo introduced a fruitful topological method which is based on the properties of the consequent mapping i e the map of w onto its boundary c u under the action of solutions of e here w is a subset of the domain of f the kneserhukuhara theorem and a classical topological result concerning continua this leads to existence results for the bvp ee g under rather general assumptions on f as well as on the boundary functions g and h in 121 by

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

classification of non degenerate projective varieties with non zero prolongation and application to target rigidity the prolongation mathfrak g k of a linear lie algebra mathfrak g subset mathfrak gl v plays an important role in the study of symmetries of g structures cartan and kobayashi nagano have given a complete classification of irreducible linear lie algebras mathfrak g subset mathfrak gl v with non zero prolongations

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

existence and uniqueness for two point boundary value problems boundary value problems associated with y f x y y for 0 x 1 are considered using techniques based on the shooting method conditions are given on f x y y which guarantee the existence on 0 1 of solutions of some initial value problems working within the class of such solutions conditions are then given on nonlinear boundary conditions of the form g y 0 y 0 0 h y 0 y 0 y 1 y 1 0 which guarantee the existence of a unique solution of the resulting boundary value problem