Topological Methods, Variational Methods and Their Applications

Preface.

Contents.

List of Academic Committees, List of Organizing Committees and Organizor.

List of Invited Speakers.

Under weak non degeneracy assumptions we show that for certain superlinear, stationary, periodic Schrödinger equations there exist infinitely many geometrically distinct two-bump solutions that change sign exactly once. We give examples of potentials that satisfy the assumptions.

We present a sequence of results about some natural generalizations of the one fixed center problem and the two fixed centers problem. The main original point is that we do not feel obliged to stay in the traditional Lagrangian or Hamiltonian frameworks.

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We present some recent works ([BW1,BW2]) of the authors on standing waves for nonlinear Schrödinger equations with a potential V.

A solution of the form ψ(x,t) = exp(−iEt/h)υ(x) is called a standing wave. There have been numerous works on the standing waves with a frequency E < inf V. Here some existence results and new features of standing waves with a critical frequency E = inf V will be illustrated.

In this paper, we study Dirichlet problem for a class of semilinear elliptic equations -Δu + λu = |u|^2*-2u + μ|u|^q-2u in Ω, , where Ω ⊂ R^N is a bounded domain, N ≥ 3, , 1 < q < 2, µ > 0, λ ∈ (–λ₁(Ω),0], λ₁(Ω) is the first eigenvalue of the Laplace operator –Δ with Dirichlet boundary conditions. By using variational method, we prove the existence of infinitely many solutions with positive energy for small µ > 0, when Ω exhibits suitable symmetry.

The heat flow is applied to replace the gradient flow for semilinear elliptic variational problems in the critical point theory. The method is also extended to study the Plateau problem for minimal surfaces.

I discuss some properties of the Eight solution of the three-body problem, many of them conjectural. I describe in particular a simple approach to the P₁₂ family, proposed by C. Marchal, which is a choreography in the rotating frame with the same 12-fold symmetry as the Eight.

We demonstrate how boundary blow-up solutions of some elliptic equations can be used to obtain better understanding of various well-known nonlinear problems on a bounded domain or over the entire R^N.

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We present a finite dimensional reduction for perturbed variational functionals and discuss some nonlinear elliptic PDE with Sobolev critical growth in bounded domains.

We discuss the p(x)–Laplacian equation of the form.

where p(x) is a function. This equation is a generalization of the p-Laplacian equation where p is a constant. In this paper we summarize our research results about the p(x)-Laplacian equations. The main contents contain: Lavrentiev phenomenon, generalized Sobolev space W^1,p(x) (Ω), regularity of weak solutions, strong maximum principle, the “first eigenvalue”, existence and multiplicity of solutions.

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A maximum principle is proved for the weak solutions u ∈ L^∞ (R×T³) of the telegraph equation u_tt − Δ_zu + cu_t + λu = f(t, x), in space dimension three, when c > 0, λ ∈ (0, c²/4] and f ∈L^∞(R × T³) (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_zu + cu_t = F(t,x,u). Applications are given to the semilinear telegraph equation with Duffing and with sine-Gordon nonlinearities.

We study nonlinear ODE problems in the complex plane, with the right hand side f(u, t) being complex analytic function of the space variable u with the coefficients being nonconstant periodic functions in the time variable t and having non-zero Fourier coefficients for non-negative indices only. Assuming first that , where h is holomorphic in the two variables, we drop out the local a priori estimate for the existence of T-periodic solutions of previous works. We prove that there exists r_o > 0, not depending of T, such that for every r < r₀ the equation has periodic solution. Next we show that the same still holds for the mentioned above class of nonlinear problems. This improves previous author results, also joint with A. Borisovich.

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We make some remarks on the method of obtaining a priori estimates for the solutions of superlinear elliptic equations having prescribed Morse index. In particular, we show that sometimes these estimates can be obtained without performing blow-up procedures.

We provide a unified proof of the asymptotics of the self-dual Maxwell-Chern-Simons vortices, as the Maxwell term is neglected, in both the U(1) and CP(1) case. This result is achieved by identifying and analyzing a suitable class of nonlinear elliptic systems with exponential type nonlinearities.

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We study the exact multiplicity of positive solutions of the p-Laplacian Dirichlet problem.

where p > 1, φ_p (y) = |y|^p−2 y, (φ_p (u′))′ is the one-dimensional p-Laplacian, and λ > 0 is a bifurcation parameter. Assuming that satisfies (F1)-(F4), we show that the bifurcation curve has exactly one critical point, a maximum, on the (‖u‖_∞, λ)-plane. Thus we are able to determine the exact multiplicity of positive solutions. We give an interesting application for a nonlinear Dirichlet problem of polynomial nonlinearities with positive coefficients.

We survey the recent results about the symmetry property of extremal functions for a family of weighted Sobolev inequalities due to Caffarelli, Kohn and Nirenberg.

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Using variational methods and Moeckel’s theorem on the saddle point property of planar central configurations in R^k(k ≥ 3), we prove the existence of a noncollision and nonplanar periodic solution for Newtonian-like N ≥ 4 body problems in R^k(k ≥ 3).

Sufficient conditions are obtained for the existence of periodic positive solution of an impulsive delay differential equation.

No abstract received.