Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth■ where s ∈(0,1),N 4 s,and λ 0 is a parameter,2_s~*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0 λ~*+∞such that the problem has exactly two positive solutions if λ∈(0,λ~*),no positive solutions for λλ~*,a unique solution(λ~*,u_(λ~*))if λ=λ~*,which shows that(λ~*,u_(λ~*)) is a turning point in H~s(R~N) for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.     

NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS

In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* O(h).     

DETECTING AND SWITCHING METHODS FOR SIMPLE BIFURCATION POINTS

For a finite-dimensional equation G(y, t)=0, where G: D(?)R~(n+1)→R~n, suppose that on its primary solution curve there is a simple bifurcation point x~*= (y~*, t~*) from where a secondary solution curve is branching off. Then during the trace process of the primary curve by a continuation method, it is always necessary to locate x~* and to find another point on the secondary curve for switching to trace it. This paper presents a proof of a practical criterion for detecting simple bifurcation points and constructs a simplified perturbation algorithm for switching branches. The convergence result of the algorithm is also given. Our experiments show that the new method is more effective than other perturbation methods, especially for large scale problems.     

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u=φ(r)up-1,u>0 in RN, u ∈ D1,2(RN),where N≥ 3,x = (x',z)∈ RK×RN-K,2≤K≤N,r =|x'|.It is proved that for 2(N -s)/(N-2) < p < 2* = 2N/(N -2),0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p=2*, the above equation does not have a ground state solution but a cylindrically symmetric-solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p,zp) and as p→2*, |x'p|→r0 which attains the maximum of φ on RN.The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

GLOBAL EXISTENCE OF THE SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear parabolic equation: where Ω is the exterior domain of a compact set in R~n with smooth boundary and F satisfies |F(λ)|=o(|λ|~2), near λ=0. It is proved that when n≥3, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property ‖u(t)‖_(L∞(Ω)=o(t~(-n/2)), as t→ ∞.     

The differentiability of solutions for elliptic equations which degenerate on part of the boundary of a convex domain

In this paper,we study the differentiability of solutions on the boundary for equartions of type L_λu=~2u/x~2+|x|~(2λ)~2u/y~2=f(x,y),where λ is an arbitrary positive number. By introducing a proper metric that is related to the elliptic operator L_λ, we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied. The main difficulty is the construction of new barrier functions in this article.     

ON THE PROPERTIES OFε(≥0) OPTIMAL POLICIES IN DISCOUNTED UNBOUNDED RETURN MODEL

This paper investigates the properties of ε(≥0) optimal policies in the model of [2].It is shownthat,if π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…)is a β-discounted optimal policy,then(π_0~*,π_1~*,…,π_n~*)~∞ for alln≥0 is also a β-discounted optimal policy.Under some condition we prove that stochastic stationarypolicy π_n~(*∞)corresponding to the decision rule π_n~* is also optimal for the same discounting factor β.Wehave also shown that for each β-optimal stochastic stationary policy π_0~(*∞),π_0~(*∞) can be decomposed intoseveral decision rules to which the corresponding stationary policies are also β-optimal separately;and conversely,a proper convex combination of these decision rules is identified with the former π_0~*.We have further proved that for any (ε,β)-optimal policy,say π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…),(π_0~*,π_1~*,…,π_(n-1)~*)∞ is ((1-β~n)~(-1)ε,β)optimal for n>0.At the end of this paper we mention that the resultsabout convex combinations and de     

关于含临界指数的p-平均曲率算子

This paper is concerned with the existence of positive solutions of the following Dirichlet problem for p-mean curvature operator with critical exponent: -div((1 |▽u|~2)(p-2)/2▽u)=λu~(p*-1) μu~(q-1),u＞0,x∈Ω, u=0,x∈■Ω, where u∈W_0~(1,p)(Ω),Ωis a bounded domain in R~N(N＞p＞1)with smooth boundary ■Ω,2＜=p＜=q＜=P~*,P~*=(Np)/(N-p),λ,P＞0.It reaches the conclusions that this problem has at least one positive solution in the different cases.It is discussed the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with critical exponent by using Nehari-type duality property firstly.As p=2,q=p,the result is correspond to that of Laplace operator.     

Differential Equations and Singular Vectors in Verma Modules over sl(n,C)

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weightλover sl(n,C).He gave a differential-operator representation of the symmetric group S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by{σ(1)|σ∈S_n}.It is known that S_n is also the Weyl group of sl(n,C)and generated by all reflections s_αwith positive rootsα.We present an explicit formula of the solution s_α(1)for every positive rootαand show directly that s_α(1)is a polynomial if and only if〈λ+ρ,α〉is a nonnegative integer.From this,we can recover a formula of singular vectors given by Malikov et al..