Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues

Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigenvalues are studied for a four-dimensional system. The existence, number, co-existence and noncoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are obtained. The bifurcation surfaces and existence regions are also given.

Lower bounds of principal eigenvalue in dimension one

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case （one-dimensional diffusions） and ends at the discrete one （birth-death processes）. Possible generalization of the results studied here is discussed at the end of the paper

The Principal Eigenvalue for Jump Processes

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented.The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.

Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal rates.We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix.Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity,respectively.We further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal rates.Finally,we apply these results to a periodic Ross-Macdonald patch model.

SEMILINEAR HEMIVARIATIONAL INEQUALITIES WITH STRONG RESONANCE AT INFINITY

A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and extend results in the literature, two results on existence are obtained.

ON NONCOERCIVE (p,q)-EQUATIONS

We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1<q<p).The reaction is(p-1)-linear near±∞and the problem is noncoercive.Using variational tools and truncation and comparison techniques together with critical groups,we produce five nontrivial smooth solutions all with sign information and ordered.In the particular case when q=2,we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.

Isoperimetric,Sobolev,and Eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method:A Survey

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

An Extension of the Hess-Kato Theorem to Elliptic Systems and Its Applications to Multiple Solution Problems

The Hess-Kato Theorem on the simplicity of the first eigenvalue of a second order elliptic operator is extended to elliptic systems.The theorem is applied to figure out the critical groups of a mountain pass point for functionals which have nonlinear ellipitic systems as Euler Lagrange Equations.A muliple solution result is obtained via the ordered Banach space method and the critical group calculations.

Spectral Gap and Logarithmic Sobolev Constant for Continuous Spin Systems

The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.

Asymptotic profile of a mutualistic model on a periodically evolving domain

In order to investigate the impact of periodically evolving domain on the mutualism interaction of two species,we study a mutualistic model on a periodically evolving domain.To overcome the difficulty caused by the advection and dilution terms,we transform the model to a reaction-difusion problem in a fixed domain.By means of eigenvalue problems,the threshold parameters are introduced.The asymptotic profiles of the solutions on an evolving domain are studied by using the threshold parameters and the upper and lower solutions method.The impact of the domain evolution rate on the persistence or extinction of species is analyzed.Numerical simulations are performed to illustrate our analytical results.

Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues.Second,we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

Bifurcation Problems for a Class of Degenerate Quasilinear Elliptic Equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.

Some Results on Subelliptic Equations

In this paper, we consider the principal on R^n, and we find a comparison principle for such semi-linear sub-elliptic equation in the whole R^n and infinitely many positive solutions of the problem. eigenvalue problem for Hormander＇s Laplacian principal eigenvalues. We also study a related prove that, under a suitable condition, we have

Limiting process of absorbing Markov chains

We outline an approach to investigate the limiting law of an absorbing Markov chain conditional on having not been absorbed for long time. The main idea is to employ Donsker-Varadhan＇s entropy functional which is typically used as the large deviation rate function for Markov processes. This approach provides an interpretation for a certain quasi-ergodicity