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Silvia Cingolani

Publications and source records attributed to Silvia Cingolani.

At least 19 recordsLinked to original sources
arXiv2026

Ground state solutions for Hartree type equations driven by superposition operators and Pohozaev Identity

We investigate Hartree-type equations driven by a nonlocal operator $\mathcal{L}_μ$, defined as a superposition of fractional Laplacians through a signed Borel measure $μ$. Under Berestycki-Lions type assumptions, we prove the existence of a Mountain Pass solution and show that its energy level coincides with the minimum on the Pohozaev manifold. We also establish the boundedness of non-negative solutions. The proof of this fact requires a careful use of the Sobolev embedding in the iterative argument and a delicate treatment of the integrals involved in the estimates, as well as a Kato-type inequality in our general setting. Finally, we establish a general Pohozaev identity for solutions under a suitable summability assumption.

math.AP↗
arXiv2026

Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-ε})u^{2_{s}^{\sharp}-2-ε} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}Ω,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}Ω,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\partialΩ,\end{aligned} \right.\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$, $0 0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the spectral fractional Laplacian. For a general domain $Ω$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $ε\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

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arXiv2026

On the de Thélin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

In this paper we prove that the smallest eigenvalue $λ_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{λ_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $λ_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

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arXiv2026

A Lagrangian approach for prescribed mass solutions of cubic-quintic Schrödinger equations and $L^2$-supercritical problems

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$): $$ (*)_m \quad - Δu + μu = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb R}^N} u^2\, dx = m,$$ where $g(s) \in C({\mathbb R},{\mathbb R})$, $m > 0$ and $μ\in {\mathbb R}$ is an unknown Lagrangian multiplier. We take an approach using a Lagrangian formulation of $(*)_m$: $$J_m(μ,u)={1\over 2}\int_{{\mathbb R}^N} |\nabla u|^2\,dx -\int_{{\mathbb R}^N} G(u)\,dx +μ\left({1\over 2}\int_{{\mathbb R}^N} u^2\, dx-m\right) \in C^1((0,\infty)\times H_r^1({\mathbb R}^N), {\mathbb R})$$ and we give new general existence results through the function: $$ b_m:\, (0,\infty) \to {\mathbb R};\ μ\mapsto \text{Mountain Pass minimax value for}\ (u\mapsto J_m(μ,u)).$$ We will show the existence of solutions of $(*)_m$ related to local minima and local maxima of $b_m(μ)$. As applications, we study cubic-quintic type equations and $L^2$-supercritical problems. In particular, when $N=2,3$, we show new existence results of normalized solutions without assuming global Ambrosetti-Rabinowitz type conditions, which partially improve the preceding results due to Jeanjean [24] and Jeanjean-Lu [26, 28].

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arXiv2025

Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -Δu=(|x|^{-(n-2)}\ast u^{p-ε})u^{p-1-ε}\quad \mbox{in}~~Ω,~~ u=0\quad \mbox{on}~~\partialΩ, \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(λ_{i,ε}, v_{i,ε})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.

math.AP↗
arXiv2025

Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent

In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-Δu=(|x|^{-{(n-2)}}\ast u^{p-ε})u^{p-1-ε} \quad \mbox{in}\quad Ω, &u>0\quad \mbox{in}\quad\hspace{1mm} Ω, &u=0\quad \mbox{on}\hspace{2.5mm}\partialΩ, \end{aligned} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is energy-critical exponent. We study the asymptotic behavior of least energy solutions as $ε\rightarrow0$. These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point $x_0$, we prove that $x_0$ is a global maximum point of the Robin's function of $Ω$.

math.AP↗
arXiv2025

Normalized ground states for NLS equations with mass critical nonlinearities

We study normalized solutions $(μ,u)\in \mathbb{R} \times H^1(\mathbb{R}^N)$ to nonlinear Schrödinger equations $$ -Δu + μu = g(u)\quad \hbox{in}\ \mathbb{R}^N, \qquad \frac{1}{2}\int_{\mathbb{R}^N} u^2 dx = m, $$ where $N\geq 2$ and the mass $m>0$ is given. Here $g$ has an $L^2$-critical growth, both at the origin and at infinity, that is $g(s)\sim |s|^{p-1}s$ as $s\sim 0$ and $s\sim\infty$, where $p=1+\frac{4}{N}$. We continue the analysis started in [Cingolani-Gallo-Ikoma-Tanaka, 2024], where we found two (possibly distinct) minimax values $\underline{b} \leq 0 \leq \overline{b}$ of the Lagrangian functional. In this paper we furnish explicit examples of $g$ satisfying $\underline{b}<0<\overline{b}$, $\underline{b}=0<\overline{b}$ and $\underline{b}<0=\overline{b}$; notice that $\underline{b}=0=\overline{b}$ in the power case $g(t)=|t|^{p-1}t$. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on $g$ to obtain the existence of a positive solution for perturbations of $g$.

math.AP↗
arXiv2025

Polyharmonic Nonlinear Scalar Field Equations

In this paper, we present a result on the existence of ground state solutions for the polyharmonic nonlinear equation $(-Δ)^m u=g(u)$, assuming that $g$ has a general subcritical growth at infinity, inspired by Berestycki and Lions \cite{BerestyckiLions}. In comparison with the biharmonic case studied in \cite{Med-Siem}, the presence of a higher-order operator gives rise to several analytical challenges, which are overcome in the present work. Furthermore, we establish a new polyharmonic logarithmic Sobolev inequality.

math.AP↗
arXiv2025

A new framework for Ljusternik-Schnirelmann theory and its application to planar Choquard equations

We consider the planar logarithmic Choquard equation $$- Δu + a(x)u + (\log|\cdot| \ast u^2)u = 0,\qquad \text{in } \mathbb{R}^2$$ in the strongly indefinite and possibly degenerate setting where no sign condition is imposed on the linear potential $a \in L^\infty(\mathbb{R}^2)$. In particular, we shall prove the existence of a sequence of high energy solutions to this problem in the case where $a$ is invariant under $\mathbb{Z}^2$-translations. The result extends to a more general $G$-equivariant setting, for which we develop a new variational approach which allows us to find critical points of Ljusternik-Schnirelmann type. In particular, our method resolves the problem that the energy functional $Φ$ associated with the logarithmic Choquard equation is only defined on a subspace $X \subset H^1(\mathbb{R}^2)$ with the property that $\|\cdot\|_X$ is not translation invariant. The new approach is based on a new $G$-equivariant version of the Cerami condition and on deformation arguments adapted to a family of suitably constructed scalar products $\langle \cdot, \cdot \rangle_u$, $u \in X$ with the $G$-equivariance property $\langle g \ast v , g \ast w \rangle_{g \ast u} = \langle v , w \rangle_u.$

math.AP↗
arXiv2025

Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity

We study the following nonlinear Schrödinger equation and we look for normalized solutions $(μ,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=|s|^{p-1}s +h(s)$, $h(s)=o(|s|^p)$ as $s\sim 0$ and $s\sim\infty$. The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s) = |s|^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}\int_{{\bf R}^N}ω_1^2\, dx$ is the mass of a least energy solution $ω_1$ of $-Δω+ω=ω^p$ in ${\bf R}^N$. We prove the existence of a positive solution for $m=m_1$ when $h$ has a sublinear growth at infinity, i.e., $h(s)=o(s)$ as $s\sim\infty$. In contrast, we show non-existence results for $h(s)\not=o(s)$ ($s\sim 0$) under a suitable monotonicity condition.

math.AP↗
arXiv2025

On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality

In this paper we consider the $N$-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$, $N \geq 2$. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N \geq 2$, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant proved by Carleson and Chang for balls in $\mathbb{R}^N$.

math.AP↗
arXiv2025

A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N

In the paper we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of estremal functions or blow-up, where the domain is the ball or the entire space. We also show that the extremal functions satisfy suitable Euler-Lagrange equations. When the domain is the entire space, such equation can be derived by N-Laplacian Schrodinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends [16] to any dimension N bigger than 2.

math.AP↗
arXiv2024

New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D

The present survey is devoted to results on Trudinger-Moser inequalities in two dimension. We give a brief overview of the history of these celebrated inequalities and, starting from the geometric problem that motivated Moser's original work, we discuss the connection between Onofri's inequality for the unit sphere and sharp inequalities on Euclidean domains. Finally, we present recent results and new insights into nonlocal interaction energy functionals in two dimension, involving logarithmic kernels.

math.AP↗
arXiv2024

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

In this paper we study the following nonlinear fractional Choquard-Pekar equation \begin{equation}\label{eq_abstract} (-Δ)^s u + μu =(I_α*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$*$} \end{equation} where $μ>0$, $s \in (0,1)$, $N \geq 2$, $α\in (0,N)$, $I_α\sim \frac{1}{|x|^{N-α}}$ is the Riesz potential, and $F$ is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ odd or even: we consider both the case $μ>0$ fixed and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed. Here we also simplify some arguments developed for $s=1$ in [Calc. Var. PDEs, 2022]. A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions [ARMA, 1983]; for \eqref{eq_abstract} the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as $μ$ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any $m>0$. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a $C^1$-regularity.

math.AP↗
arXiv2021

On fractional Schrödinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- Δ)^s u + μu = (I_α*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} in the case of general nonlinearities $F \in C^1(\mathbb{R})$ of Berestycki-Lions type, when $N \geq 2$ and $μ>0$ is fixed. Here $(-Δ)^s$, $s \in (0,1)$, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $I_α$, $α\in (0,N)$. We prove existence of ground states of \eqref{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [25, 65].

math.AP↗
arXiv2021

Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{ \begin{aligned} (-Δ)^{s} u + μu &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr \int_{\mathbb{R}^N} u^2 dx &= m, & \cr u \in H^s_r&(\mathbb{R}^N), & \end{aligned} \right. \label{problemx} \end{equation*} where $N\geq 2$, $s\in (0,1)$, $m>0$, $μ$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R}, \mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrange formulation of the problem $(P_m)$, we prove the existence of a weak solution with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev's mountain in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in [21,25]. A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd.

math.AP↗
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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