Teresa D'Aprile

Bio: Teresa D'Aprile is an academic researcher from University of Rome Tor Vergata. The author has contributed to research in topics: Dirichlet boundary condition & Dirichlet problem. The author has an hindex of 14, co-authored 43 publications receiving 902 citations.

Non-Existence Results for the Coupled Klein-Gordon-Maxwell Equations

Abstract: In this paper we obtain some non-existence results for the Klein{Gordon equation coupled with the electrostatic fleld. The method relies on the deduction of some suitable Poho‚zaev identity which provides necessary conditions to get existence ‡ ·

On Bound States Concentrating on Spheres for the Maxwell--Schrödinger Equation

Abstract: We study the semiclassical limit for the following system of Maxwell--Schrodinger equations: \[ -\frac{\hbar^2}{2m}\Delta v + v + \omega\phi v - \gamma v^{p} =0, \;\; -\Delta\phi = 4\pi\omega v^2, \] where $\hbar$, m, $\omega$, $\gamma >0$, v, $\phi: \mathbb{R}^3 \to \mathbb{R}$, $1 < p < \frac{11}{7}$. This system describes standing waves for the nonlinear Schrodinger equation interacting with the electrostatic field: the unknowns v and $\phi$ represent the \emph{wave function} associated to the particle and the electric potential, respectively. By using localized energy method, we construct a family of positive radially symmetric bound states $(v_\hbar, \phi_\hbar)$ such that $v_\hbar$ concentrates around a sphere $\{|x| = s_0\}$ when $\hbar \to 0$.

Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem

Abstract: We study the following system of Maxwell-Schrodinger equations $$ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $$ where δ > 0, u, ψ : \(\psi: {\mathbb R}^N \to {\mathbb R}\), f : \({\mathbb R} \to {\mathbb R}\), N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δK > 0 such that, for 0 < δ < δK, the system has a solution (uδ, ψδ) with the property that uδ has K spikes centered at the points \(Q_{1}^\delta,\ldots, Q_K^\delta\). Furthermore, setting \(l_\delta=\min_{i ot = j} |Q_i^\delta -Q_j^\delta|\), then, as δ → 0, \((\frac{1}{l_\delta} Q_1^\delta,\ldots, \frac{1}{l_\delta} Q_K^\delta)\) approaches an optimal configuration for the following maximization problem: $$ \max\bigg\{\sum_{{i eq j}}\frac{1}{|Q_i-Q_j|^{N-2}}\,\Big|\, (Q_i,\ldots, Q_K)\in {\mathbb R}^{NK},\, |Q_i-Q_j|\geq 1\hbox{ for }i eq j\bigg\}. $$

Existence of static solutions of the semilinear Maxwell equations

Abstract: In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$ and the magnetic field ${\bf H}$ does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ $ abla\times $ ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$ is the gauge potential related to the magnetic field ${\bf H}$ ( ${\bf H}= abla\times {\bf A}$ ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15

Multiple bound states for the Schroedinger-Poisson problem

Abstract: In this paper, we study the problem \[ \left\{\begin{array}{@{}l} -\Delta u + u + V(x)u = u^p,\\[3pt] -\Delta V = \lambda u^2, \quad \displaystyle\lim_{|x| \to +\infty} V(x)=0, \end{array}\right. \] where u, V : ℝ3 → ℝ are radial functions, λ > 0 and 1 < p < 5. We give multiplicity results, depending on p and on the parameter λ.

Non-Existence Results for the Coupled Klein-Gordon-Maxwell Equations

Abstract: In this paper we obtain some non-existence results for the Klein{Gordon equation coupled with the electrostatic fleld. The method relies on the deduction of some suitable Poho‚zaev identity which provides necessary conditions to get existence ‡ ·