Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Equimultiplicity,algebraic elimination,and blowing-up

Given a variety X over a perfect field, we study the partition defined on X by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant, instead of the Hilbert Samuel function.     

On certain multiplicity one theorems

We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4). The first-named author was partially supported by the National Security Agency (#MDA904-02-1-0020).     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Two formulas for the BR multiplicity

We prove a projection formula, expressing a relative Buchsbaum–Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula.     

Global Jacquet–Langlands correspondence,multiplicity one and classification of automorphic representations

In this paper we generalize the local Jacquet-Langlands correspondence to all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one Theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Mœglin–Waldspurger and Jacquet–Shalika for GL(n).     

Nonexistence of bounded energy solutions for a critical exponent problem

In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in R^N. The weights depend on a positive parameter γ, which is allowed to increase to infinity. While for small values of γ solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as γ increases to infinity. We also prove a multiplicity result for high energy solutions.     

Mixed multiplicities and the minimal number of generator of modules

Let (R,m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R^p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees’ mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R^p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.     

Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations

The existence of one non-trivial solution for a nonlinear problem on compact d-dimensional ( ${d \geq 3}$ ) Riemannian manifolds without boundary, is established. More precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial weak solution. Moreover, as a consequence of our approach, a multiplicity result is presented, requiring the validity of the Ambrosetti–Rabinowitz hypothesis. Successively, the Cerami compactness condition is studied in order to obtain a similar multiplicity theorem in superlinear cases. Finally, applications to Emden-Fowler type equations are presented.     

Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter β is sufficiently large.     

Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential

In this paper, we study a nonlinear elliptic problem at resonance driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions due to Chang. We prove a theorem guaranteeing the existence of one solution which is smooth and strictly positive. Then by strengthening the assumptions, we establish a multiplicity result providing the existence of at least two distinct solutions. Our hypotheses permit resonance and asymmetric behavior at +∞ and −∞. As a byproduct of our analysis we obtain an nonlinear and nonsmooth generalization of a result of Brézis–Nirenberg about H₀¹ versus C₀¹ minimizers of a smooth functional.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Bahri-Coron type theorem for the scalar curvature problem on high dimensional spheres

We consider the existence and multiplicity results for the prescribed scalar curvature problem on the standard spheres of high dimension n ?? 7. Given a C ² positive function K, using the theory of critical points at infinity, we prove an existence result as Bahri-Coron theorem. Our case is a generalization of Li (J Differ Equ 120:319?C410, 1995). Indeed, here the function K is flat near some critical points as in Li (J Differ Equ 120:319?C410, 1995) and it can have some nondegenerate critical points with ?? K ?? 0. Furthermore, using some topological arguments, we prove another kind of result.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.     

Multiple solutions for a two-point boundary value problem

In this paper, using a recent result of Ricceri, we prove two multiplicity theorems for the problem −u^″=λf(u)+μg(x,u), u(0)=u(1)=0, extending a previous result that G. Bonanno obtained for μ=0.     

Permanental roots and the star degree of a graph

For undirected graphs, without loops or multiple edges, we define the star degree of a graph, and prove that it is equal to the multiplicity of the root 1 of per(xI ? B), where B = D + A. Considering bipartite graphs, we prove that per(xI ? B) = per(xI ? L), where L = D ? A, and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per(xI ? L).     

Generic multiplicity for a scalar field equation on compact surfaces

We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.     

Spectral analysis of certain compact factors for Gaussian dynamical systems

For factors of a Gaussian automorphismT determined by compact subgroups of the group of unitary operators acting onL ² of the spectral measure ofT, we prove that the maximal spectral multiplicity is either 1 or infinity. As an application, we show that the maximal multiplicity of those factors an allL ^p, 1<p<+∞, is the same.     

Quasi-Nearly Subharmonic Functions and Quasiconformal Mappings

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mapping of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if u?°?f is quasi-nearly subharmonic for all quasi-nearly subharmonic u and f satisfying some additional conditions, then f is quasiconformal. Similar results are further established for the class of regularly oscillating functions.