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Marco Squassina

Publications and source records attributed to Marco Squassina.

At least 37 records · Page 2Linked to original sources
arXiv2023

Stein-Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein-Weiss type inequalities with general kernels on the upper half space and study the existence of extremal functions for this inequality with the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of the positive solutions of the corresponding Euler-Lagrange integral system. As an application, we finally derive some Liouville type results for the Hartree type equations in the half space.

math.AP↗
arXiv2022

Local Uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~~~\text{in}~Ω,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partialΩ, \end{cases} \end{equation*} where $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for $\varepsilon$ small.

math.AP↗
arXiv2022

Stability of variational eigenvalues for the fractional p-Laplacian

By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.

math.AP↗
arXiv2022

Concavity properties for solutions to $p$-Laplace equations with concave nonlinearities

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities.

math.AP↗
arXiv2021

Nonlocal approximations to anisotropic Sobolev norms

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent $p(x)$ reaches the value $1$.

math.AP↗
arXiv2021

An asymptotic expansion for the fractional $p$-Laplacian and for gradient dependent nonlocal operators

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the \emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) $p$-Laplacian (for $p\geq 2$) and to other gradient dependent nonlocal operators.

math.AP↗
arXiv2020

Nonlocal characterizations of variable exponent Sobolev spaces

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

math.AP↗
arXiv2019

Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.

math.AP↗
arXiv2019

Logarithmic Bose-Einstein condensates with harmonic potential

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schrödinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.

math.AP↗
arXiv2018

Modulational stability of ground states to nonlinear Kirchhoff equations

We investigate the stability of ground states to a nonlinear focusing Schrödinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground states in the spirit of one due to Weinstein [{\it SIAM J. Math. Anal.}, 16(1985),472-491].

math.AP↗
arXiv2018

Fine boundary regularity for the degenerate fractional $p$-Laplacian

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Ω$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Hölder regularity up to the boundary, that is, $u/d^s\in C^α(\overlineΩ)$ for some $α\in(0,1)$, $d$ being the distance from the boundary.

math.AP↗
arXiv2018

Nonexistence for hyperbolic problems on Riemannian manifolds

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

math.AP↗
arXiv2018

Generalized solutions of variational problems and applications

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.

math.FA↗
arXiv2018

New characterizations of Sobolev metric spaces

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.

math.AP↗