arXiv subjects

Gaetano Siciliano

Publications and source records attributed to Gaetano Siciliano.

At least 37 records · Page 2Linked to original sources

arXiv2018

The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field

We study, with respect to the parameter $q\neq0$, the following Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$ \begin{equation*} \left\{ \begin{aligned} -&Δu+ωu+q^2ϕu=|u|^{p-2}u, \\ &-Δϕ+a^2Δ^2 ϕ= 4πu^2, \end{aligned} \right. \end{equation*} where $p\in(2,3], ω>0, a\geq0$ are fixed. We prove, by means of the fibering approach, that the system has no solutions at all for large values of $q's$, and has two radial solutions for small $q's$. We give also qualitative properties about the energy level of the solutions and a variational characterization of these extremals values of $q$. Our results recover and improve some results in the literature.

math.AP↗

arXiv2018

Least energy radial sign-changing solution for the Schröinger-Poisson system in r3 under an asymptotically cubic nonlinearity

In this paper we consider the following Schrödinger-Poisson system in the whole $\mathbb R^{3}$, \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+ λϕu=f(u) &\text{ in } \mathbb R^3, -Δϕ= u^2 &\text{ in } \mathbb R^3, \end{array} \right. \end{equation*} where $λ>0$ and the nonlinearity $f$ is "asymptotically cubic" at infinity. This implies that the nonlocal term $ϕu$ and the nonlinear term $f(u)$ are, in some sense, in a strict competition. We show that the system admits a least energy sign-changing and radial solution obtained by minimizing the energy functional on the so-called {nodal Nehari set).

math.AP↗

arXiv2018

Positive solutions for Kirchhoff problems with vanishing nonlocal term

In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})Δu = f(u) & \mbox{in $Ω$,} u=0 & \mbox{on $\partialΩ$,} \end{array}\right. \end{equation*} in a bounded domain, allowing the function $m$ to vanish in many different points. Under an appropriated {\sl area condition}, by using a priori estimates, truncation techniques and variational methods, we prove a multiplicity result of positive solutions which are ordered in the $H_{0}^{1}(Ω)$-norm.

math.AP↗

arXiv2018

Quasilinear Schrödinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions

In this paper we consider the following quasilinear Schrödinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - Δu +ϕu = f(u) &\ \mbox{in } Ω, -Δϕ- \varepsilon^{4}Δ_4 ϕ= u^{2} & \ \mbox{in } Ω, u=ϕ=0 & \ \mbox{on } \partialΩ\end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, ϕ_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associated Schrödinger-Poisson system, that is by making $\varepsilon=0$ in the system above.

math.AP↗

arXiv2018

Multiplicity of positive solutions for a quasilinear Schrödinger equation with an almost critical nonlinearity

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -Δu - Δ(u^2)u = |u|^{p-2}u & \mbox{ in } Ωu= 0 &\mbox{ on } \partialΩ, \end{array} \right. \end{equation*} where $Ω$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More specifically we prove that, for $p$ near the critical exponent $22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $Ω$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.

math.AP↗

arXiv2017

On the Schrödinger-Born-Infeld system

In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrödinger and the Born-Infeld lagrangians, this latter replacing the role that, classically, is played by the Maxwell lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.

math.AP↗

arXiv2017

Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -Δ(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~Ω, u>0& \mbox{in}~Ω, u= 0& \mbox{on}~\partialΩ, \end{array} \right. \end{equation*} in a bounded and smooth domain $Ω$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.

math.AP↗

arXiv2017

Existence and asymptotic behaviour of solutions for a quasi-linear schrodinger-poisson system under a critical nonlinearity

In this paper we consider the following quasilinear Schrödinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - Δu +u+ϕu = λf(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -Δϕ-\varepsilon^{4} Δ_4 ϕ= u^{2} & \ \mbox{in } \mathbb{R}^{3}, \end{array} \right. $$ depending on the two parameters $λ,\varepsilon>0$. We first prove that, for $λ$ larger then a certain $λ^{*}>0$, there exists a solution for every $\varepsilon>0$. Later, we study the asymptotic behaviour of these solutions whenever $\varepsilon$ tends to zero, and we prove that they converge to the solution of the Schrödinger-Poisson system associated.

math.AP↗

arXiv2017

On a generalized timoshenko-kirchhoff equation

In this paper we consider a generalized fourth order nonlinear Kirchhoff equation in a bounded domain in $\mathbb R^{N}, N\geq2$ under Navier boundary conditions and with sublinear nonlinearity. We employ a change of variable which reduces the problem to a semilinear one. Then variational and topological tools are used in order to prove the existence of a solution.

math.AP↗

arXiv2016

Ground state solutions for fractional scalar field equations under a general critical nonlinearity

In this paper we study existence of ground state solution to the following problem $$ (- Δ)^αu = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^α(\mathbb R^N) $$ where $(-Δ)^α$ is the fractional Laplacian, $α\in (0,1)$. We treat both cases $N\geq2$ and $N=1$ with $α=1/2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\geq2$, exponential if $N=1$.

math.AP↗

arXiv2016

On a generalized Kirchhoff equation with sublinear nonlinearities

In this paper we consider a generalized Kirchhoff? equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools.

math.AP↗

arXiv2016

Positive semiclassical states for a fractional Schrödinger-Poisson system

We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e. in the "semiclassical limit") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.

math.AP↗

arXiv2015

A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in $\mathbb R^{N}$

In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-Δ)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$ where $0<s<1$, $(-Δ)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions and the nonlinearity $f:\mathbb R \to \mathbb R$ satisfy suitable assumptions; in particular it is assumed that $V$ achieves its positive minimum on some set $M.$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\varepsilon\to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set $M$.

math.AP↗

arXiv2014

On fractional Choquard equations

We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.

math.AP↗

arXiv2014

A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity

In this paper we give a multiplicity result for the following Chern-Simons-Schrödinger equation \[ -Δu+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s τu^2(τ) \ d τ$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.

math.AP↗

arXiv2014

Deforming solutions of geometric variational problems with varying symmetry groups

We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lacking regularity, several non-smooth extensions of the result are discussed. Among such applications is the submanifold problem of deforming the ambient metric preserving a given variational property of a prescribed family of submanifolds, e.g., constant mean curvature, up to the action of the corresponding ambient isometry groups.

math.DG↗

arXiv2013

Equivariant bifurcation in geometric variational problems

We prove an extension of a celebrated equivariant bifurcation result of J. Smoller and A. Wasserman, in an abstract framework for geometric variational problems. With this purpose, we prove a slice theorem for continuous affine actions of a (finite-dimensional) Lie group on Banach manifolds. As an application, we discuss equivariant bifurcation of constant mean curvature hypersurfaces, providing a few concrete examples and counter-examples.

math.DG↗

arXiv2013

On the equivariant implicit function theorem with low regularity and applications to geometric variational problems

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-)Riemannian geodesics, and constant mean curvature hypersurfaces.

math.DG↗