arXiv subjects

Patrizia Pucci

Publications and source records attributed to Patrizia Pucci.

At least 19 recordsLinked to original sources

arXiv2025

Uniqueness of radial solutions for $m$-Laplacian equations in low dimensions

This paper extends the uniqueness results of Serrin and Tang [\textit{Indiana Univ. Math. J.}, 49 (2000), pp. 897--923] to the low-dimensional case $1\leq N\leq m$ with $m>1$. We consider radial solutions of the overdetermined problem \[ \begin{cases} -\Delta_m u = f(u), \quad u>0 & \text{in } B_R,\\[4pt] u = \partial_\nu u = 0 & \text{on } \partial B_R, \text{ if } R<\infty,\\[4pt] \displaystyle\lim_{|x|\to\infty} u(x)=0, & \text{if } R=\infty, \end{cases} \] where $B_R$ is the open ball in $\mathbb{R}^N$ centered at the origin with radius $R>0$ (the case $R=\infty$ corresponds to the whole space, for studying positive ground states). Under suitable assumptions on the nonlinearity $f$, we establish the uniqueness of such solutions, whenever they exist. Our analysis is motivated by connections to sharp forms of the Gagliardo--Nirenberg and Nash inequalities. Although the overall framework follows that of Serrin and Tang, the details of our proofs differ substantially in the low-dimensional setting. In particular, Serrin and Tang explicitly noted that their techniques rely heavily on the condition $N>m$ and do not readily extend to $N\leq m$ (see Subsection~6.2 of their work). The present paper closes this gap, thereby providing a complete uniqueness theory for all dimensions. As a concrete example, for the canonical nonlinearity $f(u) = -u^p + u^q$ with $p m$ and $m^* = \infty$ for $N\leq m$. Consequently, our work also completely resolves an open problem posed by Pucci and Serrin [\textit{Indiana Univ. Math. J.}, 47 (1998), pp. 501--528], which had been settled for $N>m$ in the earlier work of Serrin and Tang.

math.AP↗

arXiv2025

The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system

In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3, -\Delta\phi=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which means that the Schr\"{o}dinger operator $-\Delta +V$ is allowed to be indefinite. To deal with this more difficult case, by a local linking argument and Morse theory, the system has a nontrivial solution. Furthermore, we also show the asymptotical behavior result of this solution. Additionally, the proofs rely on new observations regarding the solutions of the Poisson equation. As a main novelty with respect to corresponding results in \cite{MR4527586,MR3148130,MR2810583}, we only assume that $f$ satisfies the super-linear growth condition at the origin. We believe that the methodology developed here can be adapted to study related problems concerning the existence of solutions for Schr\"{o}dinger-Poisson system.

math.AP↗

arXiv2024

Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities

In this paper, we study {existence and multiplicity} of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left\{\begin{array}{l} -\Delta u-\Delta_q u+\lambda u=f(u) \quad x \in \mathbb{R}^N , \int_{\mathbb{R}^N}u^2 d x=c^2, \end{array}\right. \end{equation*} where $1 0$ is a constant. The nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, with mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution and the existence of a second solution with higher energy.

math.AP↗

arXiv2024

Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation}\label{Equation1} \left\{\begin{aligned} &-\Delta u-\Delta_q u+\lambda u=g(u),\quad x \in \mathbb{R}^N, &\int_{\mathbb{R}^N}u^2 d x=c^2, \end{aligned}\right. \tag{$\mathscr E_\lambda$} \end{equation} where $1 0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and the behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim \limits _{s \rightarrow 0} g(s) / s=-\infty$, which includes the logarithmic nonlinearity $$ g(s)= s \log s^2. $$ We consider a family of approximating problems that can be set in $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$ and the corresponding least-energy solutions. Then, we prove that such a family of solutions converges to a least-energy solution to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1\left(\mathbb{R}^N\right)\cap D^{1, q}\left(\mathbb{R}^N\right)$, we prove the existence of infinitely many solutions of the above $(2, q)$-Laplacian equation.

math.AP↗

arXiv2023

Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-Δ)^{s}u = λu^{-q} + u^{2^*_s-1}, \quad u>0 \quad \text{in }Ω, \mathcal A(u) = 0 \quad \text{on}~ \partialΩ= \sum_{D} \cup \sum_{\mathcal{N}}, \end{cases} \tag{$P_λ$} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partialΩ$, $1/2 0$ is a real parameter, $ 0 < q < 1 $, $N>2s$, $2^*_s=2N/(N-2s)$ and $$\mathcal{A}(u)= u \mathcal{X}_{\sum_{D}} + {\partial_νu}\mathcal{X}_{ \sum_{\mathcal{N}}}, \quad{\partial_ν=\frac{\partial }{\partialν}}.$$ Here $\sum_{D}$, $\sum_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial Ω$ such that $\sum_{D} \cup \sum_{\mathcal{N}}= \partialΩ$, $\sum_{D} \cap \sum_{\mathcal{N}}= \emptyset $ and $\sum_{D} \cap \overline{\sum_{\mathcal{N}}} = τ'$ is a smooth $(N-2)$ dimensional submanifold of $\partialΩ$. Within a suitable range of $λ$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.

math.AP↗

arXiv2023

Superlinear elliptic equations with unbalanced growth and nonlinear boundary condition

In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma.

math.AP↗

arXiv2022

Existence and multiplicity results for a class of Kirchhoff-Choquard equations with a generalized sign-changing potential

In the present work we are concerned with the following Kirchhoff-Choquard-type equation $$-M(||\nabla u||_{2}^{2})\Delta u +Q(x)u + \mu(V(|\cdot|)\ast u^2)u = f(u) \mbox{ in } \mathbb{R}^2 , $$ for $M: \mathbb{R} \rightarrow \mathbb{R}$ given by $M(t)=a+bt$, $ \mu >0 $, $ V $ a sign-changing and possible unbounded potential, $ Q $ a continuous external potential and a nonlinearity $f$ with exponential critical growth. We prove existence and multiplicity of solutions in the nondegenerate case and guarantee the existence of solutions in the degenerate case.

math.AP↗

arXiv2021

Fractional elliptic systems with critical nonlinearities

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-Δ)^s u = \fracα{2_s^*}|u|^{α-2}u|v|^β+f(x)\;\;\text{in}\;\mathbb{R}^{N}, &(-Δ)^s v = \fracβ{2_s^*}|v|^{β-2}v|u|^α+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, v >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $α,\,β>1$, $α+β=2N/(N-2s)$, and $f,\, g$ are nonnegative functionals in the dual space of $\dot{H}^s(\mathbb{R}^{N})$. When $f=0=g$, we show that the ground state solution of the above system is {\it unique}. On the other hand, when $f$ and $g$ are nontrivial nonnegative functionals with ker$(f)$=ker$(g)$, then we establish the existence of at least two different positive solutions of the above system provided that $\|f\|_{(\dot{H}^s)'}$ and $\|g\|_{(\dot{H}^s)'}$ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system.

math.AP↗

arXiv2020

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $Ω\subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

math.DG↗

arXiv2020

Fractional Hardy-Sobolev equations with nonhomogeneous terms

The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.

math.AP↗

arXiv2020

Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems

The paper deals with a nontrivial density result for $C^m(\overline{\Omega})$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(\Omega;\Gamma)= \left\{u\in W^{k,p}(\Omega): u_{|\Gamma}\in W^{\ell,p}(\Gamma)\right\},$$ endowed with the norm of $(u,u_{|\Gamma})$ in $W^{k,p}(\Omega)\times W^{\ell,p}(\Gamma)$, where $\Omega$ is a bounded open subset of ${\mathbb R}^N$, $N\ge 2$, with boundary $\Gamma$ of class $C^m$, $k\le \ell\le m$ and $1\le p<\infty$. Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in $\Omega$ and the other on $\Gamma$. Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of $\Gamma$ and, as a preliminary step, we shall prove an analogous result when either $\Omega={\mathbb R}^N$ or $\Omega={\mathbb R}^N_+$ and $\Gamma=\partial{\mathbb R}^N_+$. \keywords{Density results\and Sobolev spaces \and Smooth functions \and the Laplace--Beltrami operator.

math.AP↗

arXiv2020

Nonhomogeneous systems involving critical or subcritical nonlinearities

This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.

math.AP↗

arXiv2020

On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\geq 1$ in $\mathbb{R}^{N}$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{\dot{H}^s(\mathbb{R}^{N})'}$ is small enough (but $f\not\equiv 0$), we establish existence of at least two positive solutions to ($\mathcal E$).

math.AP↗

arXiv2019

A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.

math.AP↗

arXiv2019

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) for the parabolic problem in general bounded domains. Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior, with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions {\it generically} lose boundary conditions after GBU. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.

math.AP↗

arXiv2018

On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds

This paper is about the influence of Geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, in particular by the study of entire graphs with prescribed mean curvature, we consider classes of coercive differential inequalities of the form $$ \mathrm{div}\left( \frac{φ(|\nabla u|)}{|\nabla u|} \nabla u\right) \ge b(x)f(u) l(|\nabla u|) \qquad \text{(respectively, $\le$ or $=$)} $$ on domains of a manifold $M$, for suitable $φ,b,f,l$, with emphasis on mean curvature type operators. We investigate the validity of strong maximum principles, compact support principles and Liouville type theorems; in particular, the goal is to identify sharp thresholds, involving curvatures or volume growth of geodesic balls in $M$, to guarantee the above properties under appropriate Keller-Osserman type conditions, and to discuss the geometric reasons behind the existence of such thresholds. The paper also aims to give a unified view of recent results in the literature. The bridge with Geometry is realized by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau's Hessian and Laplacian principles and subsequent improvements.

math.AP↗

arXiv2016

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

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.

math.AP↗