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Massimo Grossi

Publications and source records attributed to Massimo Grossi.

At least 19 recordsLinked to original sources
arXiv2026

Uniqueness and nondegeneracy of positive solutions to elliptic equations with Robin boundary conditions

In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} $$ where $\beta > 0$, $p$ is subcritical, and $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.

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arXiv2026

Qualitative properties of eigenfunctions in domains with small holes

In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-\Delta$ with Dirichlet boundary condition in a smooth bounded domain $\Omega$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $\Omega_\epsilon:=\Omega\setminus B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P$ and radius $\epsilon$, for $P\in\Omega$ and $\epsilon$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$; 2) the simplicity of the eigenvalues of $-\Delta$ in $\Omega_\epsilon$; 3) the behavior of nodal sets of the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.

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arXiv2026

Lane-Emden Problems on Convex Domains of $\mathbb S^2$

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with zero Dirichlet boundary condition. We prove that for $0 \leq p < 1$ the unique positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly concave in $\Omega$, while for $1 < p \leq 3$ every positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly convex in $\Omega.$ For $p=0,$ our result gives the strict $1/2-$concavity of the torsion function in $\Omega.$ For $p=1,$ a result due to Lee and Wang gives the strict log-concavity of the first eigenfunction in $\Omega.$ As a consequence, for each $0 \leq p \leq 3,$ any positive solution has strictly convex superlevel sets and a unique nondegenerate maximum.

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arXiv2026

Bifurcations in Isoperimetric Problems with Nonlocal Interactions

We study isoperimetric problems modeled on the liquid drop model, with nonlocal interactions under a volume constraint. While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls. These new solutions lie arbitrarily close to balls and can have arbitrarily large volume. Conversely, at radii outside this sequence, no bifurcation occurs, and nearby solutions are trivial, arising only from rigid motions.

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arXiv2026

Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions

In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} \end{equation*} where $p \ge 0$, $\beta > 0$, and $\Omega$ is a bounded smooth domain. We will prove that, for all $p\ge0$, the solution $u_\beta$ behaves like a constant as $\beta\to0$. However, the value of this constant is strongly influenced by the value of $p$. Indeed, \begin{itemize} \item if $0 \le p < 1$, $u_\beta$ blows up uniformly in $\Omega$ as $\beta \to 0$. \item if $p=1$ (eigenvalue problem), $u_\beta$ converge to a constant. \item if $p>1$ $u_\beta$ converge uniformly to zero. \end{itemize} In the critical and supercritical regime $p \ge \frac{N+2}{N-2}$, the existence of solutions is no longer guaranteed a priori. In this case, when $\Omega$ is a ball and $0<\beta<\frac{2}{p-1}$ we prove the existence of a radial positive solution.

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arXiv2025

Qualitative analysis on the critical points of the Kirchhoff-Routh function

In this paper, we study the number of critical points of the Kirchhoff-Routh function \begin{equation*} \mathcal{KR}_D(x,y)=\Lambda_1^2\mathcal{R}_D(x)+\Lambda_2^2\mathcal{R}_D(y)-2\Lambda_1\Lambda_2G_D(x,y), \end{equation*} where $D$ is a bounded domain in $\mathbb{R}^2$, $x,y\in D$, $\Lambda_1,\Lambda_2>0$, $\mathcal{R}_D$ is the Robin function, and $G_D$ is the Green function of the operator $-\Delta$ with $0$ Dirichlet boundary condition on $D$. This function arises from concentration phenomena in nonlinear elliptic problems and from the de-singularization problem for the steady Euler equation. For domains with a small hole, we establish not only the exact number and the location of the critical points of $\mathcal{KR}_D$, but also their nondegeneracy. We show that the location of the hole plays a crucial role. Finally in the context of elliptic problems, we establish the existence of multiple two-peak solutions.

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arXiv2025

An extension of Cabr\'{e}-Chanillo theorem to the $p$-laplacian

In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment.

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arXiv2025

The role of the curvature of a surface in the shape of the solutions to elliptic equations

We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-\Delta u=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.

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arXiv2025

Low regularity results for degenerate Poisson problems

In this paper we study the Poisson problem, \[ \begin{cases} -{\rm div}(d^\beta\nabla u)=f&{\rm in}\ \Omega\\ u=0&{\rm on}\ \partial\Omega, \end{cases} \] where $\Omega\subset\mathbb R^N$, $N\ge2$ is a smooth bounded domain, $f$ is a continuous function, $\beta< 1$, and $d(x)=dist(x,\partial\Omega )$. We describe the behaviour of $u$ near $\partial\Omega$ and discuss some of its regularity properties.

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arXiv2024

On the critical points of solutions of Robin boundary problems

In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem \[ \begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ \partial_\nu u+\beta u=0&\text{on }\partial\Omega, \end{cases} \] where $\Omega\subseteq\mathbb{R}^2$ is a smooth and bounded domain with strictly positive curvature of the boundary, $f\ge0$ is a smooth function and $\beta>0$. Moreover, for $\beta$ large the result fails as soon as the domain is no more convex, even if it is very close to be: indeed, in this case it is possible to find solutions with an arbitrary large number of critical points.

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arXiv2023

On the critical points of solutions of PDE in a non-convex settings: the case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely $$\begin{cases} -\Delta u=\lambda e^{u}&\mbox{ in }\Omega u=0 & \mbox{ on }\partial\Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^2$ is a bounded smooth domain and $\lambda>0$ is a small parameter.

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arXiv2023

On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument.

math.DG↗
arXiv2023

On the shape of solutions to elliptic equations in possibly non convex planar domains

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.

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arXiv2022

Qualitative analysis on the critical points of the Robin function

Let $Ω\subset\mathbb{R}^N$ be a smooth bounded domain with $N\ge2$ and $Ω_ε=Ω\backslash B(P,ε)$ where $B(P,ε)$ is the ball centered at $P\inΩ$ and radius $ε$. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in $Ω_ε$ for $ε$ small enough. We will show that the location of $P$ plays a crucial role on the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green function near to $\partial B(P,ε)$. Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.

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arXiv2021

On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has \emph{large eccentricity} then the eigenfunction has \emph{exactly} two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([Jer95a]) and Grieser-Jerison ([GJ96]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [GJ09].

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arXiv2021

On the number of critical points of solutions of semilinear equations in $\mathbb{R}^2$

In this paper we construct families of bounded domains $Ω_\varepsilon$ and solutions $u_\varepsilon$ of \[\begin{cases} -Δu_\varepsilon=1&\text{ in }\ Ω_\varepsilon\\ u_\varepsilon=0&\text{ on }\ \partialΩ_\varepsilon \end{cases}\] such that, for any integer $k\ge2$, $u_\varepsilon$ admits at least $k$ maxima points for small enough $\varepsilon$. The domain $Ω_\varepsilon$ is "not far" to be convex in the sense that it is starshaped, the curvature of $\partialΩ_\varepsilon$ vanishes at exactly $two$ points and the minimum of the curvature of $\partialΩ_\varepsilon$ goes to $0$ as $\varepsilon\to0$.

math.AP↗
arXiv2021

Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -Δu=u^{p} &{\text{in}~Ω},\\[0.5mm] u>0 &{\text{in}~Ω},\\[0.5mm] u=0 &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.

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