Estimates for solutions of quasilinear problems with dead cores

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense.     

Asymptotic behaviour of large solutions of quasilinear elliptic problems

The paper deals with the large solutions of the problems $\triangle u=u^p$ and $\triangle u= e^u.$ They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P-function to establish the asymptotic behaviour of the gradients.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

Applications of pseudocanonical covers to tight closure problems

It is well known that nice conditions on the canonical module of a ring have a strong impact in the study of strong F-regularity and F-purity. For example, in the Gorenstein (or even -Gorenstein) case strong F-regularity is equivalent to weak F-regularity, and it is conjectured that this is true in general. In this note, we will show how to use the double cover of a ring to obtain sufficient conditions for strong F-regularity and F-purity. Our results involve a closure operation for a pair of ideals that could be of relevance for the conjecture mentioned above.     

An elementary construction of tilting complexes

Let A be an artin algebra and e∈A an idempotent with add(eA_A)=add(D(_AAe)). Then a projective resolution of Ae_eAe gives rise to tilting complexes for A, where P(l)^• is of term length l+1. In particular, if A is self-injective, then is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes {T(2l)^•}_l?1 for A, where T(2l)^• is of term length 2l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes {T(l)^•}_l?1 for A, where T(l)^• is of term length l+1.     

A characterization of the periods of periodic points of 1-norm nonexpansive maps

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that , where denotes the set of periods of restricted admissible arrays on 2n symbols. The important point of this equality is that is determined by arithmetical and combinatorial constraints only, and that it can be computed in finite time. By using this equality the set R(n) is computed for . Furthermore it is shown that the largest element of R(n) satisfies:      

Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds

Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich symmetries is proved by concentration compactness based on actions of the manifold's isometry group.     

Uniqueness of positive radial solutions for Δu−u+u=0 on an annulus

We prove uniqueness of positive radial solutions to the semilinear elliptic equation , subject to the Dirichlet boundary condition on an annulus in . As a by-product, our argument also provides a much simpler, if not the simplest, new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space .     

Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents

We describe special asymptotic structures of solutions of the semilinear heat equation