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Antonio Iannizzotto

Publications and source records attributed to Antonio Iannizzotto.

At least 19 recordsLinked to original sources
arXiv2026

On boundary regularity for the fractional p-Laplacian with unbounded reactions

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $Ω\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(Ω)$ for some $q\ge 1$. We prove that the unique solution $u$ is $α$-Hölder continuous up to the boundary, for any $α$ below $p'(s-N/pq)$ if $N/ps N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_Ω^s$ admits a Hölder continuous extension to the closure of $Ω$, where ${\rm d}_Ω$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

math.AP↗
arXiv2026

Integral Harnack estimates and the rate of extinction of singular fractional diffusion

We prove several integral Harnack-type inequalities for local weak solutions of parabolic equations with measurable and bounded coefficients, describing singular s-fractional p-Laplacian diffusion. Then we apply the aforementioned estimates to evaluate the decay rate of the local mass and supremum of the solutions as they approach a possible extinction time. Yet we show consistency of our general decay estimates by studying the extinction phenomenon for weak solutions of the Cauchy-Dirichlet problem, by means of an approximation procedure that carefully avoids the use of an integrable time derivative.

math.AP↗
arXiv2025

Existence of solutions for nonlinear equations with mixed local and nonlocal operators

We study an elliptic equation, with homogeneous Dirichlet boundary conditions, driven by a mixed type operator (the sum of the Laplacian and the fractional Laplacian), involving a parametric reaction and an undetermined source term. Applying a recent abstract critical point theorem of Ricceri, we prove existence of a solution for a convenient source and small enough parameters.

math.AP↗
arXiv2025

A clustering theorem in fractional Sobolev spaces

We prove a general clustering result for the fractional Sobolev space $W^{s,p}$: whenever the positivity set of a function $u$ in a square has measure bounded from below by a multiple of the cube's volume, and the $W^{s,p}$-seminorm of $u$ is bounded from above by a convenient power of the cube's side, then $u$ is positive in a universally reduced cube. Our result aims at applications in regularity theory for fractional elliptic and parabolic equations. Also, by means of suitable interpolation inequalities, we show that clustering results in $W^{1,p}$ and $BV$, respectively, can be deduced as special cases.

math.AP↗
arXiv2024

A survey on boundary regularity for the fractional p-Laplacian and its applications

We survey some recent regularity results for fractional p-Laplacian elliptic equations, especially focusing on pure and weighted boundary Hölder continuity of the solutions of related Dirichlet problems. Then, we present some applications of such results to general nonlinear elliptic equations of fractional order, treated via either variational or topological methods.

math.AP↗
arXiv2024

Optimization problems in rearrangement classes for fractional $p$-Laplacian equations

We discuss two optimization problems related to the fractional $p$-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $p$-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the maximization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution.

math.AP↗
arXiv2024

Multiple solutions for superlinear fractional $p$-Laplacian equations

We study a Dirichlet problem driven by the (degenerate or singular) fractional $p$-Laplacian and involving a $(p-1)$-superlinear reaction at infinity, not necessarily satisfying the Ambrosetti-Rabinowitz condition. Using critical point theory, truncation, and Morse theory, we prove the existence of at least three nontrivial solutions to the problem.

math.AP↗
arXiv2024

Fine boundary regularity for the singular fractional p-Laplacian

We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $Ω$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case $p\in(1,2)$ and with $s\in(0,1)$. We prove that $u/{\rm d}_Ω^s$ has a $α$-Hölder continuous extension to the closure of $Ω$, ${\rm d}_Ω(x)$ meaning the distance of $x$ from the complement of $Ω$. This result corresponds to that of ref. [28] for the degenerate case $p\ge 2$.

math.AP↗
arXiv2023

Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a (p-1)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to 'asymptotic' weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for the existence of a solution. Our work extends classical results due to Brezis-Oswald and Diaz-Saa to the quasilinear nonlocal framework.

math.AP↗
arXiv2023

Positive solutions for the fractional p-Laplacian via mixed topological and variational methods

We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We assume local conditions ensuring the existence of sub- and supersolutions. So we prove existence of two positive solutions, in both the coercive and noncoercive cases.

math.AP↗
arXiv2022

Bifurcation-type results for the fractional p-Laplacian with parametric nonlinear reaction

We study a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction, which depends on a positive parameter. The reaction is assumed to be (p-1)-sublinear near the origin and (p-1)-superlinear at infinity (including the concave-convex case). Following a variational approach based on a combination of critical point theory and suitable truncation techniques, we prove a bifurcation-type result for the existence of positive solutions.

math.AP↗
arXiv2022

Monotonicity of eigenvalues of the fractional p-Laplacian with singular weights

We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.

math.AP↗
arXiv2021

Five solutions for the fractional p-Laplacian with noncoercive energy

We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. Thus we focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably truncated reactions, we show the existence of five nontrivial solutions: two positive, two negative, and one nodal.

math.AP↗
arXiv2021

Four solutions for fractional p-Laplacian equations with asymmetric reactions

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.

math.AP↗
arXiv2021

Multiple solutions for the fractional p-Laplacian with jumping reactions

We study a nonlinear elliptic equation driven by the degenerate fractional p-Laplacian, with Dirichlet type condition and a jumping reaction, i.e., (p-1)-linear both at infinity and at zero but with different slopes crossing the principal eigenvalue. Under two different sets of hypotheses, entailing different types of asymmetry, we prove the existence of at least two nontrivial solutions. Our method is based on degree theory for monotone operators and nonlinear fractional spectral theory.

math.AP↗
arXiv2021

On the logistic equation for the fractional p-Laplacian

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.

math.AP↗
arXiv2020

Existence and multiplicity of positive solutions for the fractional Laplacian under subcritical or critical growth

We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under a (explicitly determined) threshold. As an application, we find two positive solutions for a fractional Brezis-Nirenberg problem.

math.AP↗