Let $E=\{u\in C^1[0,1]:u\left(0\right)=u\left(1\right)=0\}$. Let $S_k^{\nu }$ with $\nu =\{+,-\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^{\nu }$-solutions of the problem $$\left\{\begin{array}{cc}{\left({\displaystyle \frac{u^{\text{'}}}{\sqrt{1-u^{\text{'}2}}}}\right)}^{\text{'}}+\lambda a\left(x\right)f\left(u\right)=0,\hfill & x\in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0,\hfill \end{array}\right.$$ where $\lambda >0$ is a parameter, $a\in C\left(\right[0,1],(0,\infty \left)\right)$. We determine the intervals of parameter $\lambda$ in which the above problem has one, two or three $S_k^{\nu }$-solutions. The proofs of the main results are based upon the bifurcation technique.

We prove Fefferman's SAK Principle for a class of hypoelliptic operators on R2 whose nonnegative symbol vanishes anisotropically on the characteristic manifold.

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...

In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions ${\chi }_a(t,x)$ form a family of evolving regular curves in ${ℝ}^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of ${\chi }_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...

We study the decay in time of solutions of a symmetric regularized-long-wave equation and we show that under some restriction on the form of nonlinearity, the solutions of the nonlinear equation have the same long time behavior as those of the linear equation. This behavior allows us to establish a nonlinear scattering result for small perturbations.

We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure ⎧uₜₜ - Δu = -F₁(|u|²,|v|²)u, ⎨ ⎩vₜₜ - Δv = -F₂(|u|²,|v|²)v, for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential...

We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in ${ℝ}^n$. The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

We give some theorems on continuity and differentiability with respect to (h,t) of the solution of a second order evolution problem with parameter $h\in \Omega \subset {ℝ}^m$. Our main tool is the theory of strongly continuous cosine families of linear operators in Banach spaces.