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Angela Pistoia

Publications and source records attributed to Angela Pistoia.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the critical points of Steklov eigenfunctions

We consider the critical points of Steklov eigenfunctions on a compact, smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For generic metrics on $M$ we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to $\partial M$, and the Euler characteristic of $M$. In dimension $2$ this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of $M$ and of the number of sign changes of $u$ on $\partial M$. In the case of the second Steklov eigenfunction on a genus $0$ surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on $M$ Steklov eigenfunctions are Morse functions in $M$.

math.AP↗
arXiv2024

Bubble solution for the critical Hartree equation in pierced domain

In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -Δu=\left(\int_{Ω_\varepsilon}\frac{u^{2_μ^*}}{|x-y|^μ}dy\right)u^{2_μ^*-1}, &\text{ in } Ω_\varepsilon, \\ u=0, &\text{ on } \partialΩ_\varepsilon, \end{cases} \end{align*} where $2_μ^*=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\geq 5$, $0<μ<4$ with $μ$ sufficiently close to $0$, $Ω_\varepsilon:=Ω\backslash B(0,\varepsilon)$ and $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, which contains the origin, and $\varepsilon$ is a positive parameter. As $\varepsilon$ goes to zero, we construct bubble solution which blows up at the origin.

math.AP↗
arXiv2024

On a critical Hamiltonian system with Neumann boundary conditions

We consider the Hamiltonian system with Neumann boundary conditions: \[ -Δu + μu=v^{q }, \quad -Δv+ μv=u^{p} \quad \text{ in $Ω$}, \qquad u, v >0 \quad \text{ in $Ω$,} \qquad \partial_νu= \partial_νv=0 \quad \text{ on $\partial Ω$, } \] where $μ>0$ is a parameter and $Ω$ is a smooth bounded domain in $\mathbb R^N .$ When $(p, q)$ approaches from below the critical hyperbola $N/(p+1) + N/(q+1)=N-2$, we build a solution which blows-up at a boundary point where the mean curvature achieves its minimum and negative value.

math.AP↗
arXiv2024

Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator

Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14].

math.SP↗
arXiv2024

Partially concentrating standing waves for weakly coupled Schrödinger systems

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.

math.AP↗
arXiv2024

A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

We consider the eigenvalues problem for the the fractional Laplacian in a bounded domain Omega with Dirichlet boundary condition. A recent result by Fall, Ghimenti, Micheletti and Pistoia (CVPDE (2023)) states that under generic small perturbations of the coefficient of the equation or of the domain Omega all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity 2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension $2$ in C^1(Omega).

math.AP↗
arXiv2023

Positive solutions of elliptic systems with superlinear terms on the critical hyperbola

We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use the dual method on Orlicz spaces, showing that our problem possesses a mountain pass type solution.

math.AP↗
arXiv2023

Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs

We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to $3$. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than $1$. The proof is based on a suitable Ljapunov-Schmidt reduction.

math.AP↗
arXiv2023

Prescribing nearly constant curvatures on balls

In this paper we address two boundary cases of the classical Kazdan-Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of R^2, and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature. Using a Ljapunov-Schmidt procedure, we obtain new existence results when the prescribed functions are close to constants.

math.AP↗
arXiv2023

The Brezis-Nirenberg problem in 4D

The problem \begin{equation} \label{bn} -Δu=|u|^{4\over n-2}u+λV u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ \end{equation} where $Ω$ is a bounded regular domain in $\mathbb R^n$, $λ\in \mathbb R$ and $V\in C^0(\overline Ω),$ that was introduced by Brezis and Nirenberg in their famous paper, where they address the existence of positive solutions in the autonomous case, i.e. the potential $V$ is constant. Since then, a huge amount of work has been done. In the following we will make a brief history highlighting the results which are much closer to the problem we wish to study in the present paper.

math.AP↗
arXiv2023

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

We study the following Lane-Emden system \[ -Δu=|v|^{q-1}v \quad \text{ in } Ω, \qquad -Δv=|u|^{p-1}u \quad \text{ in } Ω, \qquad u_ν=v_ν=0 \quad \text{ on } \partial Ω, \] with $Ω$ a bounded regular domain of $\mathbb{R}^N$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $Ω$ is a ball or an annulus.

math.AP↗
arXiv2023

Generic properties of eigenvalues of the fractional Laplacian

We consider the Dirichlet eigenvalues of the fractional Laplacian $(-Δ)^s$, with $s\in (0,1)$, related to a smooth bounded domain $Ω$. We prove that there exists an arbitrarily small perturbation $\tildeΩ=(I+ψ)(Ω)$ of the original domain such that all Dirichlet eigenvalues of the fractional Laplacian associated to $\tildeΩ$ are simple. As a consequence we obtain that all Dirichlet eigenvalues of the fractional Laplacian on an interval are simple. In addition, we prove that for a generic choice of parameters all the eigenvalues of some non-local operators are also simple.

math.AP↗
arXiv2023

Non-uniqueness for the nonlocal Liouville equation in $\mathbb{R}$ and applications

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-Δ)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon κ(x)$ with $\varepsilon \in (0,1)$ small and $κ\in C^{1,α}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $α> 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.

math.AP↗
arXiv2023

Existence of solutions to a slightly supercritical pure Neumann problem

We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric domains (e.g. ellipsoids) are also discussed.

math.AP↗
arXiv2023

Pinwheel solutions to Schrödinger systems

We establish the existence of positive segregated solutions for competitive nonlinear Schrödinger systems in the presence of an external trapping potential, which have the property that each component is obtained from the previous one by a rotation, and we study their behavior as the forces of interaction become very small or very large. As a consequence, we obtain optimal partitions for the Schrödinger equation by sets that are linearly isometric to each other.

math.AP↗
arXiv2022

Non-degeneracy of critical points of the squared norm of the second fundamental form on manifolds with minimal boundary

Let $(M,\bar g)$ be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on $\partial M$. We show that for a generic Riemannian metric $\bar g$, the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate. We show that the generality of this property holds when we restrict ourselves to the conformal class of the initial metric on $M$.

math.AP↗