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Alessio Fiscella

Publications and source records attributed to Alessio Fiscella.

At least 19 recordsLinked to original sources
arXiv2025

Critical double phase problems involving sandwich-type nonlinearities

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad \text{in } \Omega,\quad u= 0 \quad\text{on } \partial \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overline{\Omega})$, $\lambda$, $\theta$ are real parameters, $w$ is a suitable weight and $B\colon \overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $\lambda$ and $\theta$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$.

math.AP↗
arXiv2024

Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-\Delta)^{s_{1}}_{p}u +(-\Delta)^{s_{2}}_{q}u=\lambda P(x)|u|^{k-2}u+\theta|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, \Omega,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus \Omega, $$ where $\Omega\subseteq\mathbb{R}^{N}$ is a general open set, $0 0$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ is the critical exponent. We prove that there exists a decreasing sequence $\{\theta_j\}_j$ such that for any $j\in\mathbb N$ and with $\theta\in(0,\theta_j)$, there exist $\lambda_*$, $\lambda^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $\lambda\in (\lambda_*,\lambda^*)$. On the other hand, we show there exists $\overline{\lambda}>0$ such that for any $\lambda>\overline{\lambda}$, there exists $\theta^*=\theta^*(\lambda)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta\in(0,\theta^*)$.

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arXiv2023

Mixed local-nonlocal quasilinear problems with critical nonlinearities

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on } \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb{R}^N$ is a bounded open set with smooth boundary, dimension $N\geq 2$, parameter $\lambda>0$, exponents $0<s<1<p<N$, while $q\in(1,p^{\ast})$ with $p^{\ast}=\frac{Np}{N-p}$. The problem is driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian. We analyze three different scenarios depending on exponent $q$. For this, we combine variational methods with some topological techniques, such as the Krasnoselskii genus and the Lusternik-Schnirelman category theories.

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arXiv2022

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^\mu}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<\mu<N$, $1<p<N$, $p<q<p+ \frac{\alpha p}{N}$, $0 \leq a(\cdot)\in C^{0,\alpha}(\mathbb{R}^N)$ with $\alpha \in (0,1]$ and $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

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arXiv2021

Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

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arXiv2021

On double phase Kirchhoff problems with singular nonlinearity

In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the non-degenerate as well as the degenerate Kirchhoff case in our setting.

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arXiv2021

On a class of critical double phase problems

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on $\partial\Omega$, where $\Omega \subset \mathbb{R}^N$ is a bounded Lipschitz domain, $1<\vartheta 0$, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of $\lambda^*>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $\lambda\in (0,\lambda^*)$.

math.AP↗
arXiv2021

Singular Finsler double phase problems with nonlinear boundary condition

In this paper we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

math.AP↗
arXiv2020

Existence and multiplicity results for Kirchhoff type problems on a double phase setting

In this paper, we study two classes of Kirchhoff type problems set on a double phase framework. That is, the functional space where finding solutions coincides with the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(Ω)$, with modular function $\mathcal H$ related to the so called double phase operator. Via a variational approach, we provide existence and multiplicity results.

math.AP↗
arXiv2020

A double phase problem involving Hardy potentials

In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)= γ\left(\displaystyle\frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle\frac{|u|^{q-2}u}{|x|^q}\right)+f(x,u) & \mbox{in } Ω,\\ u=0 & \mbox{in } \partialΩ, \end{array} \right. $$ where $Ω\subset\mathbb R^N$ is an open, bounded set with Lipschitz boundary, $0\inΩ$, $N\geq2$, $1<p<q<N$, weight $a(\cdot)\geq0$, $γ$ is a real parameter and $f$ is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(Ω)$, with modular function $\mathcal H(t,x)=t^p+a(x)t^q$. For this, we first introduce the Hardy inequalities for space $W^{1,\mathcal H}_0(Ω)$, under suitable assumptions on $a(\cdot)$.

math.AP↗
arXiv2020

Existence of at least $k$ solutions to a fractional $p$-Kirchhoff problem involving singularity and critical exponent

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s} u&=\frac{\lambda}{|u|^{\gamma-1}u}+|u|^{p_s^*-2}u~\text{in}~\Omega,\nonumber u&>0~\text{in}~\Omega,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber \end{align} where $\Omega\subset\mathbb{R}^N$, is a bounded domain with Lipschitz boundary, $\lambda>0$, $N>ps$, $0<s,\gamma<1$, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator for $1<p<\infty$ and $p_s^*=\frac{Np}{N-ps}$ is the critical Sobolev exponent. We employ a {\it cut-off} argument to obtain the existence of $k$ (being an arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove a uniform $L^{\infty}({\Omega})$ bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using the symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.

math.AP↗
arXiv2017

A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-Δ)^s u = \displaystyle\fracλ{u^γ}+u^{2^*_s-1}&\quad\mbox{in } Ω,\\ u>0&\quad\mbox{in } Ω,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminusΩ, \end{array}\right. $$ where $Ω$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $λ>0$ is a real parameter, exponent $γ\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-Δ)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

math.AP↗
arXiv2016

Multiplicity results for fractional Laplace problems with critical growth

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-Δ)^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases} (-Δ)^su=γ|u|^{2^*-2}u+f(x,u) & \mbox{in } Ωu=0 & \mbox{in } \mathbb R^n\setminus Ω, \end{cases}$$ where $Ω\subset\mathbb R^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in(0,1)$, $γ$ is a positive real parameter, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $f$ is a Carathéodory function satisfying different subcritical conditions.

math.AP↗
arXiv2015

Gevrey regularity for integro-differential operators

We prove for some singular kernels $K(x,y)$ that viscosity solutions of the integro-differential equation $\int_{\mathbb{R}^n} \left[u(x+y)+u(x-y)-2u(x)\right]\,K(x,y)dy=f(x)$ locally belong to some Gevrey class if so does $f$. The fractional Laplacian equation is included in this framework as a special case.

math.AP↗
arXiv2014

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

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