Tiancheng Ouyang

Exact Multiplicity Results for Boundary Value Problems with Nonlinearities Generalizing Cubic

Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type The first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

An exact multiplicity result for a class of semilinear equations

For a class of Dirichlet problems in two dimensions, generalizing the model case we show existence of a critical so that there are exactly 0, 1 or 2 nontrivial solutions (in fact, positive), depending on whether We show that all solutions lie on a single smooth solution curve, and study some properties of this curve. We use bifurcation approach. The Curial thing is to show that any nontrivial solution of the corresponding linearized problem is of one sign.

Multiplicity results for two classes of boundary-value problems

Multiplicity results are provided for two classes of boundary-value problems with cubic nonlinearities, depending on a parameter

Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem

\lambda

Periodic solutions with singularities in two dimensions in the

. In particular, it is proved that for sufficiently large ...

n

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability with respect to collinear and symmetric perturbations.

-body problem

Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated.

Perturbation of Global Solution Curves for Semilinear Problems

Abstract We revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in [6]. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approach, and study the evolution of turning points. We give conditions under which the critical (turning) points continue on smooth curves, which allows us to reduce the problem to the easier case of f (0) = 0. We show that the smallest root of f (u) does not have to be restricted.

Solution curves for two classes of boundary-value problems

For a given value of a real parameter A we are interested in multiplicity of solutions, and how solutions change with A. Typically we prove that all solutions of (1.1) lie on a single solution curve. This fact is important for computation of solutions. It means that all solutions of (1.1) can be computed by very efficient continuation algorithms, and we can start the continuation of solutions somewhere on a stable branch, where it is easy to compute the solution. We also obtain exact multiplicity results for large A, and sometimes for all i. We recall our strategy in [ 1, 21. We assumed the interval (a, b) to be symmetric about origin, taking (a, 6) = (-1, 1) without loss of generality, andf(x, u) to be even in u. Under an additional condition xf, < 0 for x # 0 (see Lemma 2.2 for the precise statement) we proved that any solution of (1.1) is even. This allowed us to prove that any nontrivial solution of the variational problem

Positivity for the linearized problem for semilinear equations

Using recent results of M. Tang [ Uniqueness of positive radial solutions for