Existence of solutions for elliptic equations with discontinuous nonlinearities strong resonance at infinity

In the present paper, our main purposes are to study nonlinear elliptic equations with strong resonance at infinity. Some existence theorems for nontrivial solutions are obtained by using some nonsmooth critical point theorems in [N. C. Kourogenis, N. S. Papageorgiou, Nonsmooth critical point theory and Nonlinear elliptic equations at resonance, J. Austral. Math Soc. (Ser. A) 69 (2000) 245–271]. The two of our theorems generalize Theorems 0.1 and 5.2 in [P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. TMA 7 (1983) 981–1012] to nonsmooth cases. Another theorem is new even if for the smooth case.     

Resonance problems for the p-Laplacian systems

Two existence theorems of the solutions are obtained for the p-Laplacian systems at resonance under a Landesman-Lazer-type condition by critical point theory.     

Existence results for obstacle problems for nonlinear hemivariational inequality at resonance

In this paper, we are interested in studying the existence of solutions to obstacle problems for nonlinear hemivariational inequality at resonance driven by the p$p$-Laplacian. Using a variational approach based on the nonsmooth critical point theory for nondifferentiable functionals. We prove two existence theorems.     

Nontrivial solutions of semilinear equations at resonance

In this paper we prove new existence results concerning nontrivial solutions to semilinear elliptic problem at resonance. The methods used here are based on combining the minimax methods and the Morse theory.     

Renormalization and blow up for the critical Yang-Mills problem

We consider the Yangs-Mills equations in 4+1 dimensions. This is the energy critical case and we show that it admits a family of solutions which blow up in finite time. They are obtained by the spherically symmetric ansatz in the SO(4) gauge group and result by rescaling of the instanton solution. The rescaling is done via a prescribed rate which in this case is a modification of the self-similar rate by a power of |logt|. The powers themselves take any value exceeding 3/2 and thus form a continuum of distinct rates leading to blow-up. The methods are related to the authors' previous work on wave maps and the energy critical semi-linear equation. However, in contrast to these equations, the linearized Yang-Mills operator (around an instanton) exhibits a zero energy eigenvalue rather than a resonance. This turns out to have far-reaching consequences, amongst which are a completely different family of rates leading to blow-up (logarithmic rather than polynomial corrections to the self-similar rate).     

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.     

Existence of solutions for a class of noncooperative elliptic systems

We establish the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of superlinear growth. Moreover, if there is resonance, we also find at least a solution. All results are obtained by the minimax methods in critical point theory.     

S^1*S^1-INDEX THEORY ON PRODUCT SPACE

A new index is constructed by use of the canonical representation of S1 x S1 group over a product space. This index satisfies the general properties of the usual index but does not satifsy the dimension property. As an application, two abstract critical point theorems are given.     

{\tf S}$^{\bold 1}${\td X}{\tf S}$^{\bold 1}$-INDEX THEORY ON PRODUCT SPACE

A new index is constructed by use of the canonical representation of $S^1\times S^1$ group over a product space. This index satisfies the general properties of the usual index but does not satifsy the dimension property. As an application, two abstract critical point theorems are given.     

A note on the critical ideals of a cycle

Critical ideals of a graph G which are the determinantal ideals of its generalized Laplacian matrix were first introduced by Corrales and Valencia as a generalization of the critical group. Then it was shown that critical ideals are also closely related to other properties of the graph, such as the clique number and the zero forcing number. In this note, we give a simple proof of Theorem 4.13 proved in [7], which gives a Gröbner basis of the first nontrivial critical ideal of a cycle. After that as applications we determine explicit expressions for the characteristic ideals of a cycle and the critical groups of a family of thick wheels.     

Nonlinear biharmonic equations with Hardy potential and critical parameter

Some embedding inequalities in Hardy-Sobolev spaces with weighted function α|x| are proved. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained. Next, we study the existence of nontrivial solutions of biharmonic equations with Hardy potential and critical parameter.     

Critical rationalism in practice: Strategies to manage subjectivity in OR investigations

The philosophical position referred to as critical rationalism (CR) is potentially important to OR because it holds out the possibility of supporting OR’s claim to offer managers a scientifically ‘rational’ approach. However, as developed by Karl Popper, and subsequently extended by David Miller, CR can only support practice (deciding what to do, how to act) in a very limited way; concentrating on the critical application of deductive logic, the crucial role of subjective judgements in making technical and moral choices are ignored or are at least left underdeveloped. By reflecting on the way that managers, engineers, administrators and other professionals take decisions in practice, three strategies are identified for handling the inevitable subjectivity in practical decision-making. It is argued that these three strategies can be understood as attempts to emulate the scientific process of achieving intersubjective consensus, a process inherent in CR. The perspective developed in the paper provides practitioners with a way of understanding their clients’ approach to decision-making and holds out the possibility of making coherent the claim that they are offering advice on how to apply a scientific approach to decision-making; it presents academics with some philosophical challenges and some new avenues for research.     

Nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups

In this paper, we study the existence of nontrivial solutions for a class of second-order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and Morse theory.     

乘积空间上的Z_2×Z_2-指标理论

本文针对Ｚ２×Ｚ２群在乘积空间上的自然作用定义一种新的指标．这种指标满足通常指标的一般性质，但不满足维数性质．作为这个指标的应用，我们还给出了两个抽象的临界点定理．     

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

Nonlinear Biharmonic Equations with Critical Potential

In this paper, we study two semilinear singular biharmonic equations： one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile, we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

Indecomposability graph and critical vertices of an indecomposable graph

Given a directed graph G=(V,A), the induced subgraph of G by a subset X of V is denoted by G[X]. A subset X of V is an interval of G provided that for a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For instance, 0?, V and {x}, x∈V, are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial, otherwise it is decomposable. Given an indecomposable directed graph G=(V,A), a vertex x of G is critical if G[V?{x}] is decomposable. An indecomposable directed graph is critical when all its vertices are critical. With each indecomposable directed graph G=(V,A) is associated its indecomposability directed graph defined on V by: given x≠y∈V, (x,y) is an arc of if G[V?{x,y}] is indecomposable. All the results follow from the study of the connected components of the indecomposability directed graph. First, we prove: if G is an indecomposable directed graph, which admits at least two non critical vertices, then there is x∈V such that G[V?{x}] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs G which have a unique non critical vertex x and such that G[V?{x}] is critical. Third, we propose a new approach to characterize the critical directed graphs.     

Nontrivial periodic solutions for asymptotically linear resonant difference problem

In this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained. The methods used here are based on combining the minimax methods and the Morse theory, especially the observation on the critical groups.     

含临界位势与临界参数的超线性椭圆型方程

本文考虑一类含临界位势与临界参数的超线性椭圆型方程解的存在性.本文应用Morse理论,考虑非线性项f(x,s)在零点附近以及无穷远处的性质,给出了方程在某个新的Sobolev-Hardy空间中解的存在性.