The Existence of Solutions for a Class of Schr¨odinger Equations via Morse Index Theory

In this paper,with the relative Morse index,we will study the existence of solutions of(1.1)under the assumptions that V satisfies some weaker conditions than those in[2].

Finite Morse Index Solutions of a Nonlinear Schr?dinger Equation

We prove Liouville type theorems for stable and finite Morse index H_(loc)^(1)∩L_(loc)^(∞)solutions of the nonlinear Schrodinger equation -Δu+λ|x|^(a)u=|x|b|^(u)|^(p-1)u in R^(N),where N≥2,λ>0,a,b>-2 and p>1,Our analysis reveals that all stable solutions of the equation must be zero for all p>1,Furthermore,finite Morse index solutions must be zero if N≥3 an p≥(N+2+2b)/(N-2).The main tools we use are integral estimates,a Pohozaev type identity and a monotonicity formula.

The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.

Sub-harmonicity,monotonicity formula and finite Morse index solutions of an elliptic equation with negative exponent

A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

The multiple solutions for one-dimensional cubic nonlinear problem u'+u^3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ'_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k.

Morse Index of Multiple Blow-Up Solutions to the Two-Dimensional Sinh-Poisson Equation

In this paper we consider the Dirichlet problem■where p is a small parameter andΩis a C^(2)bounded domain in R^(2).In[1],the author proves the existence of a m-point blow-up solution u_(p)jointly with its asymptotic behaviour.We compute the Morse index of up in terms of the Morse index of the associated Hamilton function of this problem.In addition,we give an asymptotic estimate for the first 4m eigenvalues and eigenfunctions.

LIOUVILLE THEOREM FOR CHOQUARD EQUATION WITH FINITE MORSE INDICES

In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose that 2 〈 p 〈2N-α/N-2,we will show that this problem does not possess nontrivial solution with finite Morse index. While for p =2N-α/N-2,if i（u） 〈∞, then we have ∫RN∫RN｜u（x）｜p｜u（y）｜p dxdy 〈∞ and ∫RN｜△u｜2 dx=｜∫RN∫RN｜u（x）｜p/｜x-y｜a dxdy.

MORSE CONCAVITY FOR CLOSED GEODESICS

In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.

NONTRIVIAL SOLUTIONS FOR SEMILINEAR SCHRDINGER EQUATIONS

The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.

Multiplicity of solutions for the semilinear subelliptic Dirichlet problem

In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))satisfying the Hormander's condition,f(x,u)∈C(Ω×R),g(x,u)is a Carathéodory function onΩ×R,andΩis an open bounded domain in R~n with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.

MASLOV-TYPE INDEX THEORYFOR SYMPLECTIC PATHS AND SPECTRAL FLOW(II)

Based on the spectral now and the stratification structures of the symplectic group Sp(2n, c) , the Maslov-type index theory and its generalization, the w-index theory parameterized by all ω on the unit circle, for arbitrary paths in Sp(2n, C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n, C) is proved, and the mean index for any path in Sp(2n, C) is defined. Also, the relation among various Maslov-type index theories is studied.

Iterated index formulae for closed geodesies with applications

In this paper, various precise iteration equalities and inequalities of Morse indices for the closed geodesics are established. As applications of these formulae, multiplicity results of closed geodesics on some Riemannian manifolds are proved.

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW (Ⅰ)

The spectral flow for paths of admissible operators in arbitrary Banach space is defined, and some properties of the spectral flow is studied.

Maslov-type index and brake orbits in nonlinear Hamiltonian systems

In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.

Liouville Type Results for a p-Laplace Equation with Negative Exponent

Positive entire solutions of the equation where 1 〈 p ≤ N, q 〉 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solution that has finite Morse index when q 〉 qc but it admits a family of stable positive radial entire solutions when 0 〈 q ≤ qc- Proof of the stability of positive radial entire solutions of the equation when 1 〈 p 〈 2 and 0 〈 q ≤ qc relies on Caffarelli-Kohn Nirenberg＇s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 〈 p ≤ N and q 〉 qc. The case of 1 〈 p 〈 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.

Minimal P-symmetric period problem of first-order autonomous Hamiltonian systems

Let P C Sp（2n） satisfying pk = I2n. We consider the minimal P-symmetric period problem of the autonomous nonlinear Hamiltonian system x（t） = JH＇（x（t））. For some symplectic matrices P, we show that for any π 〉0,the above Hamiltonian system possesses a kT periodic solution x with kT being its minimal P-symmetric period provided H satisfies Rabinowitz＇s conditions on the minimal period conjecture, together with that H is convex and H（Px） = H（x）.

Indices and Stability of the Lagrangian System on Riemannian Manifold

In this paper,let m≥1 be an integer,M be an m-dimensional compact Riemannian manifold.Firstly the linearized Poincare map of the Lagrangian system at critical point x d/dt L_(q)(t,x,x)−L_(p)(t,x,x)=0 is explicitly given,then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index,finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

Some Abstract Critical Point Theorems for Self-adjoint Operator Equations and Applications

By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent,the authors discuss the existence and multiplicity of solutions for(nonlinear) operator equations,and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.

Qualitative properties and classification of solutions to elliptic equations with Stein-Weiss type convolution part

In this paper,we study the qualitative properties and classification of the solutions to the elliptic equations with Stein-Weiss type convolution part.Firstly,we study the qualitative properties,such as the symmetry,regularity and asymptotic behavior of the positive solutions.Secondly,we classify the non-positive solutions by proving some Liouville type theorems for the finite Morse index solutions and stable solutions to the nonlocal elliptic equations with double weights.

On the Regularity for Stationary Harmonic Maps

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.