Simone Secchi

Università degli Studi di Milano-Bicocca | UNIMIB · Department of Mathematics and its Applications

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation $$ (-\Delta)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $...

We study the semilinear equation $-\Delta_g u + V(\sigma) u = f(u)$ on a Cartan-Hadamard manifold ${\cal M}$ of dimension $N \geq 3$, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function $V \in C({\cal M})$. In particular, the decay of $V$ at infinity is allowed, with some restrictions related to...

We consider a smooth, complete and non-compact Riemannian manifold (M,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {M},g)$$\end{document} of dimension d≥...

This chapter presents a straightforward approach to abstract measure theory. We will see that an abstract Lebesgue integral can be defined after introducing the idea of measuring sets. In the end we will connect these two approaches to integration theory.

The standard approach to the Lebesgue integral is via measure theory: we must define a set function—called a measure—on a set of suitable sets—called measurable sets, then we can define measurable functions, and finally integrable functions. The main advantage of this approach is that at the end we have the highest generality. On the other hand, su...

We study the equation −Δgw+w=λα(σ)f(w) on a d-dimensional homogeneous Cartan-Hadamard Manifold M with d≥3. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities f which have an oscillating behavior either at zero or at infinity.

We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.

We study the equation $-\Delta_g w+w=\lambda \alpha(\sigma) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infi...

We are interested in a general Choquard equation −Δ+m2u−mu+V(x)u−μ|x|u=∫RNF(y,u(y))|x−y|N−αdyf(x,u)−K(x)|u|q−2u under suitable assumptions on the bounded potential V and on the nonlinearity f. Our analysis extends recent results by the second and third author on the problem with μ=0 and pure-power nonlinearity f(x,u)=|u|p−2u. We show that, under ap...

We consider a smooth, complete and non-compact Riemannian manifold $(\mathcal{M},g)$ of dimension $d \geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -\Delta_g w + V w = \alpha f(w) + \lambda w \quad\hbox{in $\mathcal{M}$}. $$ The potential $V \colon \mathcal{M} \to \mathbb{R}$ is a continuous function which is coe...

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the Sobolev embedding sense. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some r...

We investigate the existence of solutions to the fractional nonlinear Schrödinger equation (−Δ)su=f(u)−μu with prescribed L2-norm ∫RN|u|2dx=m in the Sobolev space Hs(RN). Under fairly general assumptions on the nonlinearity f, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of so...

We show that ground state solutions to the nonlinear, fractional problem \begin{equation*} \begin{cases} (-\Delta)^{s} u + V(x) u = f(x,u) & \text{in } \Omega, \\ u = 0 & \text{in } \R^N \setminus \Omega, \end{cases} \end{equation*} on a bounded domain $\Omega \subset \R^N$, converge (along a subsequence) in $L^2 (\Omega)$, under suitable condition...

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the bounded potential \(V\) and on the nonlinearity \(f\). Our ana...

We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$We show that ground state solutions converge (along a subsequence) in \(L^2_{\mathrm {loc}} (\mathbb {R}^N)\), under suitable conditions on f and V, to a ground state solution of the local problem as \(s \...

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ i...

We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity...

We study a perturbed Schrödinger equation in the plane arising from the coupling of quantum physics with Newtonian gravitation. We obtain some existence results by means of a perturbation technique in Critical Point Theory.

We consider the nonlinear fractional problem \begin{align*} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \hbox{in $\mathbb{R}^N$} \end{align*} We show that ground state solutions converge (along a subsequence) in $L^2_{\mathrm{loc}} (\mathbb{R}^N)$, under suitable conditions on $f$ and $V$, to a \emph{very weak} solution of the local problem as $s \to...

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ \Omega, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus \Omega, \end{array} \right. \end{align*} on a bounded domain $\Omega \subset \mathbb{R}^N$, converge (along a subs...

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutio...

We survey some recent results about the Nonlinear Schrödinger Equation on a metric graph, obtained in collaboration with D. Noja and S. Rolando. We also provide some general information about some analytic tools that are useful to set a convenient environment.

We consider the semilinear fractional equation $ (I-\Delta)^s u = a(x) |u|^{p-2}u$ in $\mathbb{R}^N$, where $N \geq 3$, $0<s<1$, $2<p<2N/(N-2s)$ and $a$ is a bounded weight function. Without assuming that $a$ has an asymptotic profile at infinity, we prove the existence of a ground state solution.

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is dec...

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in R N − ∆ + m 2 u − mu + V(x)u = R N |u(y)| p |x − y| N−α dy |u| p−2 u − Γ (x)|u| q−2 u, where m > 0 and the potential V is decomposed as the sum of a Z N-periodic term and of a bounded term that decays at infinity. The result is proved by variation...

We consider the magnetic pseudo-relativistic Schrödinger equation where , m > 0, is an external continuous scalar potential, is a continuous vector potential and is a convolution kernel, is a constant, , . We assume that A and V are symmetric with respect to a closed subgroup G of the group of orthogonal linear transformations of . If for any , the...

In this paper we discuss the existence and non--existence of weak solutions to parametric equations involving the Laplace-Beltrami operator $\Delta_g$ in a complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold $(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and intrinsic metric $d_g$. Namely, our simple model is...

We consider the semilinear fractional equation (I − ∆)su = a(x)|u|p−2u in RN, where N ≥ 3, 0 < s < 1, 2 < p < 2N/(N − 2s) and a is a bounded weight function. Without assuming that a has an asymptotic profile at infinity, we prove the existence of a ground state solution.

Under different assumptions on the potential functions b and c, we study the fractional equation I-Δαu=λb(x)|u|p-2u+c(x)|u|q-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docu...

We prove that the generalized pseudorelativistic equation $$ \left( -c^2 \Delta + m^{2} c^{\frac{2}{1-s}} \right)^{s} u - m^{2s} c^{\frac{2s}{1-s}} u +\mu u = |u|^{p-1} u $$ can be solved for large values of the "light speed" $c$ even when $p$ crosses the critical value for the fractional Sobolev embedding.

We study a boundary value problem related to the search of standing waves for the nonlinear Schrödinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the double-bridge graph, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-c...

We prove the existence of a solution to the semirelativistic Hartree equation under suitable growth assumption on the potential functions

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 \Delta + m^2}u + V u = (I_\alpha * |u|^{p}) |u|^{p-2}u$ 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_\alpha (x) = \frac{c_{N,\alpha}}{|x|^{N-\alph...

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -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...

We prove that the critical problem for the fractional Laplacian in an annular type domain admits a nontrivial solution provided that the inner hole is sufficiently small.

We prove existence and multiplicity of symmetric solutions for the Schrödinger-Newton system in three-dimensional space using equivariant Morse theory.

We study the problem (-εi∇+A(x))²u+V(x)u=ε⁻²((1/(|x|))∗|u|²)u, u∈L²(R³,C), ε∇u+iAu∈L²(R³,C³), where A:R³→R³ is an exterior magnetic potential, V:R³→R is an exterior electric potential, and ε is a small positive number. If A=0 and ε=ℏ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose to describ...

We investigate the soliton dynamics for the fractional nonlinear Schrodinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.

We prove existence and multiplicity of symmetric solutions for the \emph{Schr\"odinger-Newton system} in three dimensional space using equivariant Morse theory.

We construct solutions to a class of Schrödinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.

In this note we prove the existence of radially symmetric solutions for a class of fractional Schr\"odinger equation in (\mathbb{R}^N) of the form {equation*} \slap u + V(x) u = g(u), {equation*} where the nonlinearity $g$ does not satisfy the usual Ambrosetti-Rabinowitz condition. Our approach is variational in nature, and leans on a Pohozaev iden...

By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.

We construct solutions to a class of Schr\"{o}dinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\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$,$ $N\geq3$, $\alpha\in(0,N)$ and $2-(\alpha/N) <p<(2N-\alpha)/(N-2)$. \ We assume that bo...

This work is devoted to the Dirichlet problem for the equation (-\Delta u = \lambda u + |x|^\alpha |u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $\lambda$ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided $\alpha$ is small enough. This solution has a variational characteriz...

The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of t...

By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the $p$-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required. Comment: 16 pages

We prove the existence of solutions for the singularly perturbed Schr\"odinger--Newton system {ll} \hbar^2 \Delta \psi - V(x) \psi + U \psi =0 \hbar^2 \Delta U + 4\pi \gamma |\psi|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $\psi$ concentrates, as $\hbar \to 0$, around (str...

We prove the existence of solutions for the singularly perturbed Schr\"odinger--Newton system {ll} \hbar^2 \Delta \psi - V(x) \psi + U \psi =0 \hbar^2 \Delta U + 4\pi \gamma |\psi|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $\psi$ concentrates, as $\hbar \to 0$, around (str...

In this work we consider the magnetic NLS equation $$ ( \frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb{R}^N\qquad\qquad(0.1)$$ where $N \geq 3$, $A \colon \mathbb{R}^N \to \mathbb{R}^N$ is a magnetic potential, possibly unbounded, $V \colon \mathbb{R}^N \to \mathbb{R}$ is a multi-well electric potential,...

We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity. Comment: 12 pages, accepted version

The semi-classical regime of standing wave solutions of a Schr\"odinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the...

For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.

We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories un...

We prove the existence of a positive radial solution for the H\'enon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.

We prove existence results for complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory, developed in A. Ambrosetti and M. Badiale [Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 6,...

We investigate some asymptotic properties of extrema to a two-dimensional variational problem in the unit disk. Some results about non-radialicity of solutions are given.

We prove existence of standing wave solutions for a nonlinear Schrödinger equation on 3 under the influence of an external magnetic field B. In particular we deal with the physically meaningful case of a constant magnetic field B = (0,0,b) having source in the potential A(x) = (b/2)(−x2,x1,0) corresponding to the Lorentz gauge.

The main purpose of this paper is to study the existence of single-peaked positive solutions of the singularly perturbed elliptic equation -ε2div(J(x)∇u)+V(x)u=f(u)inRN,where J is a symmetric uniformly elliptic matrix and V is a positive potential, possibly unbounded from above. If f(u)=up, then solutions concentrate at non-degenerate critical poin...

We consider the standing wave solutions of the three dimensional semilinear Schrodinger equation with competing potential functions $V$ and $K$ and under the action of an external electromagnetic vector field $A$. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. I...

In this paper we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima of an auxiliary function depending only on the potentials.

By exploiting a variational identity of Poho?zaev-Pucci-Serrin type for solutions of class C1, we get some necessary conditions for locating the peak-points of a class of singularly perturbed quasilinear elliptic problems in divergence form. More precisely, we show that the points where the concentration occurs, in general, must belong to what we c...

We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics. To cite this article: S. Secchi et al., C. R. Acad. Sci. Paris, S...

We prove the existence of one or more solutions to a singularly perturbed elliptic problema with two potential functions.

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics.

We present some results in the analysis of non-compact differential equations on unbounded domains.

We study the existence of standing waves for a class of nonlinear Schrödinger equations in , with both an electric and a magnetic field. Under suitable non-degeneracy assumptions on the critical points of an auxiliary function related to the electric field, we prove the existence and the multiplicity of complex-valued solutions in the semiclassical...

The main results give hypotheses ensuring that a non-autonomous first order Hamiltonian system has a global branch of homoclinic solutions bifurcating from an eigenvalue of odd multiplicity of the linearization. The system is required to be asymptotically periodic (as time goes to plus and minus infinity) and these limit problems should have no hom...

We prove some multiplicity results by means of a perturbation technique in critical point theory.

We prove some multiplicity results for a nonlinear equation of Schroedinger type with potential functions

We prove the existence of multiple closed geodesics on non-compact cylindrica manifolds.

We study NLS equation in presence of a magnetic field, having source in a vector potential, possibly unbounded on R, 2