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Gaetano Siciliano

Publications and source records attributed to Gaetano Siciliano.

At least 19 recordsLinked to original sources
arXiv2026

On a zero-mass $(p,q)$-Laplacian equation involving subcritical and supercritical growth in $\mathbb R^N$

This paper is concerned with the zero-mass $(p,q)$-Laplacian equation $$ -Δ_p u-Δ_q u = |u|^{r-2}u+λ|u|^{s-2}u \quad \text{in } \mathbb{R}^N, $$ where $1<q<p<N$. The exponents $r$ and $s$ may belong to either the subcritical or the supercritical range with respect to the critical Sobolev exponents $p^*$ and $q^*$. We establish existence and nonexistence results and show that the sign of the parameter $λ$ determines the solvability regimes of the equation. The existence proofs rely on variational methods, truncation arguments, and regularity theory, while the nonexistence results are derived from a suitable Pohozaev-type identity for the $(p,q)$-Laplacian operator.

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arXiv2026

On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u\\ -Δϕ+a^2Δ^2ϕ=4πu^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.

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arXiv2026

Boosted Ground States for a Pseudo-Relativistic Schrödinger Equation with a double power nonlinearity

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $ψ(t,x)=e^{iλt}φ\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t ψ=(\sqrt{-Δ+m^2})ψ- |ψ|^{\frac{2}{N}}ψ-μ|ψ|^{q}ψ~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $λ$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

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arXiv2025

Normalized solutions for Schrödinger-Bopp-Podolsky systems in bounded domains

We consider an elliptic system of Schrödinger-Bopp-Podolsky type in a bounded and smooth domain of R3 with a non constant coupling factor. This kind of system has been introduced in the mathematical literature in [14] and in the last years many contributions appeared. In particular here we present the results in [2] and [34] which show existence of solutions by means of the Ljusternik-Schnirelmann theory under different boundary conditions on the electrostatic potential.

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arXiv2025

Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour

We consider the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases} - Δu+ u+Q(x)ϕu= \vert u\vert^4 u+ λK(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip - Δϕ+ a^{2}Δ^{2} ϕ= 4πQ(x) u^{2}& \mbox{ in }\ \mathbb{R}^3. \end{cases} \end{equation*} Here $u,ϕ:\mathbb {R}^{3}\rightarrow \mathbb{R}$ are the unknowns, $Q$ and $K$ are given functions satisfying mild assumptions, $a\geq0, λ>0$ are parameters and $p\in (0,1)$. We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter $λ$ is small. Then we give general results concerning the structure of the set of solutions. We show also the behaviour of the solutions as the parameters $a,λ$ tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as $a$ tends to zero.

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arXiv2025

Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

Given a smooth bounded domain $Ω\subset \mathbb R^3$, we consider the following nonlinear Schrödinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -Δu+ ϕu -\abs{u}^{p-2}u = ωu & \quad \text{in } λΩ, -Δϕ=u^{2}& \quad \text{in }λΩ, u>0 &\quad \text{in }λΩ, u =ϕ=0 &\quad \text{on }\partial (λΩ), \int_{λΩ}u^{2} \,\text{d} x=ρ^2 \end{array} \right. \end{equation*} in the expanding domain $λΩ\subset \mathbb R^{3}, λ>1$ and $p\in (2,3)$, in the unknowns $(u,ϕ,ω)$. We show that, for arbitrary large values of the expanding parameter $λ$ and arbitrary small values of the mass $ρ>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $λΩ$. Moreover we show that as $λ\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$.

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arXiv2023

Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system

In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 Δu + (V + ϕ) u = u |u|^{p - 1} \newline - Δϕ- βΔ_4 ϕ= u^2, \end{cases} $$ where $1 < p < 5, β> 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,ϕ:\mathbb R^{3}\to \mathbb R$ in the semiclassical regime, namely when $\varepsilon\to 0.$ By means of the Lyapunov--Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential $V$.

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arXiv2022

Positive Solutions For a Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$

We consider the following Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} Δu + V(x)u + ϕu = f(u)\\ -\varepsilon^{2} Δϕ+ \varepsilon^{4} Δ^{2}ϕ= 4π\varepsilon u^{2}\\ \end{array} \right.$$ where $\varepsilon > 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R}, f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy suitable assumptions. By using variational methods, we prove that the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of $M$, the set of minima of the potential $V$.

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arXiv2022

Existence and limit behavior of least energy solutions to constrained Schrödinger-Bopp-Podolsky systems in $\mathbb{R}^3$

Consider the following Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -Δu + ωu + ϕu = u|u|^{p-2},\newline -Δϕ+ a^2Δ^2ϕ=4πu^2,\newline \|u\|_{L^2}=ρ, \end{cases} \] where $a,ρ>0$ and our unknowns are $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}^3$ and $ω\in\mathbb{R}$. We prove that if $2 0$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $2 0$ is sufficiently small, then least energy solutions are radially symmetric up to translation and as $a\to 0$, they converge to a least energy solution of the Schrödinger-Poisson-Slater system under the same $L^2$-norm constraint.

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arXiv2022

Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results

We look for critical points with prescribed energy for the family of even functionals $Φ_μ=I_1-μI_2$, where $I_1,I_2$ are $C^1$ functionals on a Banach space $X$, and $μ\in \mathbb{R}$. For several classes of $Φ_μ$ we prove the existence of infinitely many couples $(μ_{n,c}, u_{n,c})$ such that $$Φ'_{μ_{n,c}}(\pm u_{n,c}) = 0 \quad \mbox{and} \quad Φ_{μ_{n,c}}( \pm u_{n,c}) = c \quad \forall n \in \mathbb{N}.$$ More generally, we analyze the structure of the solution set of the problem $$Φ_μ'(u)=0, \quad Φ_μ(u)=c$$ with respect to $μ$ and $c$. In particular, we show that the maps $c \mapsto μ_{n,c}$ are continuous, which gives rise to a family of {\it energy curves} for this problem. The analysis of these curves provide us with several bifurcation and multiplicity type results, which are then applied to some elliptic problems. Our approach is based on the {\it nonlinear generalized Rayleigh quotient} method developed in \cite{I1}.

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arXiv2020

Multiple solutions for some strongly degenerate second order elliptic equations

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $Ω$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.

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arXiv2020

A perturbation approach for the Schrödinger-Born-Infeld system: solutions in the subcritical and critical case

In this paper, we study the following Schrödinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+ϕu=f(u)+μ|u|^4u\,\,&\mbox{in}\,\,\R^3,\\ -\textrm{div}\displaystyle\bigg(\frac{\nablaϕ}{\sqrt{1-|\nablaϕ|^2}}\bigg)=u^2&\mbox{in}\,\,\R^3,\\ u(x)\rightarrow0,\,\,ϕ(x)\rightarrow0,&\,\text{as}\,\,x\rightarrow\infty, \end{array} \right. $$ where $μ\geq0$ and $f\in C(\R,\R)$ satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case $f(u)=|u|^{p-1}u$ for $p\in(2,{5}/{2}]$ and $μ=0$ which was left in \cite{Azzollini19} as an open problem.

math.AP↗
arXiv2020

Positive solutions for a class of nonlocal problems with possibly singular nonlinearity

We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.

math.AP↗
arXiv2020

A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations

Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a noncompact group. In $\mathbb R^n$, a compactness result for invariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has been proved under the condition that the $G$ action on $\mathbb R^n$ is compatible, see \cite{willem}. As a first result we generalize this and show here that the compactness is recovered for particular subgroups of the isometry group of a Riemannian manifold. We investigate also isometric action on Hadamard manifold $(M,g)$ proving that a large class of subgroups of $\mathrm{Iso}(M,g)$ is compatible. As an application we get a compactness result for ``invariant'' functions which allows us to prove the existence of nonradial solutions for a classical scalar equation and for a nonlocal fractional equation on $\mathbb R^n$ for $n=3$ and $n=5$, improving some results known in the literature. Finally, we prove the existence of nonradial invariant functions such that a compactness result holds for some symmetric spaces of non compact type.

math.AP↗
arXiv2018

Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

We study the following nonlinear Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu + ωu + q^{2}ϕu = |u|^{p-2}u -Δϕ+ a^2 Δ^2 ϕ= 4πu^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,ω>0$. We prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schrödinger-Poisson system as $a\to0$.

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