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Silvia Cingolani

Publications and source records attributed to Silvia Cingolani.

34 records · Page 2Linked to original sources
arXiv2020

On the fractional NLS equation and the effects of the potential well's topology

In this paper we consider the fractional nonlinear Schrödinger equation $$\varepsilon^{2s}(-Δ)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N$$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is a nonlinearity satisfying Berestycki-Lions type conditions. For $\varepsilon>0$ small, we prove the existence of at least $\rm{cupl}(K)+1$ positive solutions, where $K$ is a set of local minima in a bounded potential well and $\rm{cupl}(K)$ denotes the cup-length of $K$. By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of $K$ for $\varepsilon$ small.

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arXiv2020

Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$

We perform a semiclassical analysis for the planar Schrödinger-Poisson system \[ \cases{ -\varepsilon^{2} Δψ+V(x)ψ= E(x) ψ\quad \text{in $\mathbb{R}^2$},\cr -ΔE= |ψ|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge \rightarrow 0$.

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arXiv2019

Stationary waves with prescribed $L^2$-norm for the planar Schrödinger-Poisson system

The paper deals with the existence of standing wave solutions for the Schrödinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely $$ \left \{ \begin{aligned} - Δu & + λu + γ\Bigl(\log {| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\ &\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right. $$ where $c>0$ is a given real number. Under different assumptions on $γ\in \mathbb R$, $a \in \mathbb R$, $p>2$, we prove several existence and multiplicity results. Here $λ\in \mathbb R $ appears as a Lagrange parameter and is part of the unknowns. With respect to the related higher dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer and it forces the implementation of new ideas to catch the normalized solutions.

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arXiv2017

Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation: $$ -\varepsilon^2Δv+V(x)v = \frac{1}{\varepsilon^α}(I_α*F(v))f(v) \quad \hbox{in}\ \mathbb{R}^N, $$ where $N\geq 3$, $α\in (0,N)$, $I_α(x)={A_α\over |x|^{N-α}}$ is the Riesz potential, $F\in C^1(\mathbb{R},\mathbb{R})$, $F'(s) = f(s)$ and $\varepsilon>0$ is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\varepsilon\to 0$, to a local minima of $V(x)$ under general conditions on $F(s)$. Our result is new also for $f(s)=|s|^{p-2}s$ and applicable for $p\in (\frac{N+α}{N}, \frac{N+α}{N-2})$. Especially, we can give the existence result for locally sublinear case $p\in (\frac{N+α}{N}, 2)$, which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least $\hbox{cupl}(K)+1$ solutions concentrating around $K$ as $\varepsilon\to 0$, where $K\subset Ω$ is the set of minima of $V(x)$ in a bounded potential well $Ω$, that is, $m_0 \equiv \inf_{x\in Ω} V(x) < \inf_{x\in \partialΩ}V(x)$ and $K=\{x\inΩ;\, V(x)=m_0\}$.

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arXiv2016

The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).

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arXiv2016

Amann-Zehnder type results for p-Laplace problems

The existence of a nontrivial solution is proved for a class of quasilinear elliptic equations involving, as principal part, either the p-Laplace operator or the operator related to the p-area functional, and a nonlinearity with p-linear growth at infinity. To this aim, Morse theory techniques are combined with critical groups estimates.

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arXiv2016

Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schrödinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 Δ-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $Ω\subset R^N$ such that \[ m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K = \{ x \in Ω\ | \ V(x) = m_0\}$. For $\hbar >0$ small we prove the existence of at least $cuplenght(K) + 1$ geometrically distinct, complex-valued solutions whose modula concentrate around $K$ as $\hbar \to 0$.

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arXiv2015

Nonlinear Schr{ö}dinger equation: concentration on circles driven by an external magnetic field

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schrödinger equation \begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in \mathbb{R}^{3},\end{align}where $p\textgreater{}2$, $A$ is a vector potential associated to a given magnetic field $B$, i.e $\nabla \times A =B$ and $V$ is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinity or outside a compact set.We assume that $A$ and $V$ satisfy a cylindrical symmetry. By a refined penalization argument, we prove the existence of semiclassical cylindrically symmetric solutions of upper equation whose moduli concentrate, as $\hbar \to 0$, around a circle. We emphasize that the concentration is driven by the magnetic and the electric potentials. Our result thus shows that in the semiclassical limit, the magnetic field also influences the location of the solutions of $(\ref{eq:initialabstract})$ if their concentration occurs around a locus, not a single point.

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arXiv2015

Semiclassical analysis for pseudo-relativistic Hartree equations

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 Δ+ m^2}u + V u = (I_α* |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_α(x) = \frac{c_{N,α}}{|x|^{N-α}}$ is a convolution kernel, $c_{N,α}$ is a positive constant and $(N-1)p-N<α<N$. For $N=3$, $α=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.

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arXiv2014

Ground states for the pseudo-relativistic Hartree equation with external potential

We prove existence of positive ground state solutions to the pseudo-relativistic Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-Δ+m^2} u +Vu = \left( W * |u|^θ \right)|u|^{θ-2} u \quad\text{in $\mathbb{R}^N$}\\ u \in H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m >0$, $V$ is a bounded external scalar potential and $W$ is a convolution potential, radially symmetric, satisfying suitable assumptions. We also furnish some asymptotic decay estimates of the found solutions.

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arXiv2013

Multiplicity of positive solutions of nonlinear Schrödinger équations concentrating at a potential well

We consider singularly perturbed nonlinear Schrödinger equations \be \label{eq:0.1} - \varepsilon^2 Δu + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $Ω\subset \R^N$ such that \[m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K = \{x \in Ω\ | \ V(x) = m_0\}$. For $\e >0$ small we prove the existence of at least ${\cuplength}(K) + 1$ solutions to (\ref{eq:0.1}) concentrating, as $\e \to 0$ around $K$. We remark that, under our assumptions of $f$, the search of solutions to (\ref{eq:0.1}) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

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arXiv2011

Intertwining semiclassical solutions to a Schrödinger-Newton system

We study the problem (-ε\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=ε^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}ε\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magnetic potential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exterior electric potential, and $ε$ is a small positive number. If A=0 and $ε=\hbar$ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose in \cite{pe2}\ to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that $A$ and $V$ are compatible with the action of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for any given homomorphism $τ:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential $V$ on the number of semiclassical solutions $u:\mathbb{R}% ^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=τ(g)u(x)$ for all $g\in G$, $x\in\mathbb{R}^{3}$. We also study the concentration behavior of these solutions as $ε\rightarrow0.\medskip$

math.AP↗
arXiv2011

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^α}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\ $is a real valued vector potential, $V$ is a real valued scalar potential$,$ $N\geq3$, $α\in(0,N)$ and $2-(α/N) <p<(2N-α)/(N-2)$. \ We assume that both $A$ and $V$ are compatible with the action of some group $G$ of linear isometries of $\mathbb{R}^{N}$. We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition \[ u(gx)=τ(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N}, \] where $τ:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.

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arXiv2008

Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \R^N \to \C$, under conditions on the nonlinearity which are nearly optimal.

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