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Massimo Grossi

Publications and source records attributed to Massimo Grossi.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the number of critical points of stable solutions in bounded strip-like domains

In this paper we show that there exists a family of domains $Ω_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, such that the $stable$ solution of the problem \[ \begin{cases} -Δu= g(u)&\hbox{in }Ω_\varepsilon\\ u>0&\hbox{in }Ω_\varepsilon\\ u=0&\hbox{on }\partialΩ_\varepsilon \end{cases} \] admits $k$ critical points with $k\ge2$. Moreover the sets $Ω_\varepsilon's$ are star-shaped and "close" to a strip as $\varepsilon\to0$. Next, if $g(u)\equiv1$ and $N\ge3$ we exhibit a family of domain $Ω_\varepsilon's$ with $positive$ $mean$ $curvature$ and solutions $u_\varepsilon $ which have $k$ critical points with $k\ge2$. In this case, the domains $Ω_\varepsilon $ turn out to be "close" to a cylinder as $\varepsilon\to0$.

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arXiv2020

A complete scenario on nodal radial solutions to the Brezis Nirenberg problem in low dimensions

In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -Δu=|u|^{2^*-2}u+λu&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball of $\R^N$. We compute the asymptotics of the solution $u$ as well as $||u||_\infty$, its first zero and other relevant quantities as $ł$ goes to a critical value $\barł$. Also the sign of $ł-\barł$ is established in all cases. This completes an analogous analysis for $N\ge7$ given in [EGPV].

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arXiv2020

Uniqueness of the critical point for semi-stable solution in $\mathbb{R}^2$

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.

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arXiv2020

On the number and location of critical points of solutions of nonlinear elliptic equations in domains with a small hole

In this paper we study the following problem \begin{equation} \begin{cases} -Δu=f(u)~&\mbox{in}\ Ω_\varepsilon,\\ u>0~&\mbox{in}\ Ω_\varepsilon,\\ u=0~&\mbox{on}\ \partialΩ_\varepsilon, \end{cases} \end{equation} where $Ω_\varepsilon=Ω\backslash B(P,\varepsilon)$, $Ω\subset R^N$ with $N\geq 2$ is a smooth bounded domain, $B(P,\varepsilon)$ is the ball centered at $P$ and radius $\varepsilon>0$ and $f$ is a smooth nonlinearity. By some computations involving the Green function and degree theory, we compute the number and location of critical points of solutions for small $\varepsilon>0$.

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arXiv2019

Non-uniqueness of blowing-up solutions to the Gelfand problem

We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined Lyapunov-Schmidt reduction and the computation of the degree of a finite-dimensional map.

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arXiv2019

Bubbling nodal solutions for a large perturbation of the Moser-Trudinger equation on planar domains

In this work we study the existence of nodal solutions for the problem $$ -Δu = λu e^{u^2+|u|^p} \text{ in }Ω, \; u = 0 \text{ on }\partial Ω, $$ where $Ω\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $Ω$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $Ω$ is an arbitrary domain and $λ$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser-Trudinger critical equation on a non-symmetric domain.

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arXiv2018

Sharp concentration estimates near criticality for radial sign-changing solutions of Dirichlet and Neumann problems

We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.

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arXiv2018

Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains

We compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases} -Δu= u^p & \text{in $Ω$} \\ u=0 & \text{on $\partialΩ$} \\ u>0 & \text{in $Ω$.} \end{cases} \end{equation} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large. When $Ω$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \cite{Sirakov}, gives the uniqueness of the solution to \eqref{abstr}, for $p$ large. This proves, in dimension two and for $p$ large, a conjecture by Gidas-Ni-Nirenberg [29].

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arXiv2018

Asymptotic analysis and energy quantization for the Lane-Emden problem in dimension two

We complete the study of the asymptotic behavior, as $p\rightarrow +\infty$, of the positive solutions to \[ \left\{\begin{array}{lr}-Δu= u^p & \mbox{in}Ω\\ u=0 &\mbox{on}\partial Ω\end{array}\right. \] when $Ω$ is any smooth bounded domain in $\mathbb R^2$, started in [4]. In particular we show quantization of the energy to multiples of $8πe$ and prove convergence to $\sqrt{e}$ of the $L^{\infty}$-norm, thus confirming the conjecture made in [4].

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arXiv2017

Blow-up analysis for nodal radial solutions in Moser-Trudinger critical equations in $R^2$

In this paper we consider nodal radial solutions $u_ε$ to the problem \[ \begin{cases} -Δu=λue^{u^2+|u|^{1+ε}}&\text{ in }B,\\ u=0&\text{ on }\partial B. \end{cases} \] and we study their asymptotic behaviour as $ε\searrow0$, $ε>0$. We show that when $u_ε$ has $k$ interior zeros, it exhibits a multiple blow-up behaviour in the first $k$ nodal sets while it converges to the least energy solution of the problem with $ε=0$ in the $(k+1)$-th one. We also prove that in each concentration set, with an appropriate scaling, $u_ε$ converges to the solution of the classical Liouville problem in $R^2$.

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arXiv2017

Nonradial entire solutions for Liouville systems

We consider the following system of Liouville equations: $$\left\{\begin{array}{ll}-Δu_1=2e^{u_1}+μe^{u_2}&\text{in }\mathbb R^2\\-Δu_2=μe^{u_1}+2e^{u_2}&\text{in }\mathbb R^2\\\int_{\mathbb R^2}e^{u_1}<+\infty,\int_{\mathbb R^2}e^{u_2}<+\infty\end{array}\right.$$ We show existence of at least $n-\left[\frac{n}3\right]$ global branches of nonradial solutions bifurcating from $u_1(x)=u_2(x)=U(x)=\log\frac{64}{(2+μ)\left(8+|x|^2\right)^2}$ at the values $μ=-2\frac{n^2+n-2}{n^2+n+2}$ for any $n\in\mathbb N$.

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arXiv2016

Entire radial and nonradial solutions for systems with critical growth

In this paper we establish existence of radial and nonradial solutions to the system $$ \begin{array}{ll} -Δu_1 = F_1(u_1,u_2) &\text{in }\mathbb{R}^N,\newline -Δu_2 = F_2(u_1,u_2) &\text{in }\mathbb{R}^N,\newline u_1\geq 0,\ u_2\geq 0 &\text{in }\mathbb{R}^N,\newline u_1,u_2\in D^{1,2}(\mathbb{R}^N), \end{array} $$ where $F_1,F_2$ are nonlinearities with critical behavior.

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arXiv2016

Nodal solutions for Lane-Emden problems in almost-annular domains

In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -Δu = |u|^{p-1} u \qquad & \text{in} Ω, \\ u= 0 & \text{on} \partialΩ, \end{array} \right. $$ where $Ω$ is a bounded domain in ${\mathbb R}^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1 1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as $p\to 1$. Keywords: semilinear elliptic equations, nodal solutions, supercritical problems.

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arXiv2016

A non-variational system involving the critical Sobolev exponent. The radial case

In this paper we consider the non-variational system $$ \begin{cases} -Δu_i = \sum\limits_{j=1}^k a_{ij} u_j^{(N+2)/(N-2)} &\text{in }\mathbb R^N,\newline u_i>0 &\text{in }\mathbb R^N,\newline u_i\in D^{1,2}(\mathbb R^N). \end{cases} $$ and we give some sufficient conditions on the matrix $(a_{ij})_{i,j=1,\dotsc ,k}$ which ensure the existence of solutions bifurcating from the bubble of the critical Sobolev equation.

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arXiv2015

On the Hardy-Sobolev equation

In this paper we study the Hardy problem in R^N with N>2 and in a ball B of R^N. Using a suitable map we transform the Hardy problem into another one without the singular term. Then we obtain some bifurcation results from the radial positive solutions corresponding to some explicit values of the parameter lambda.

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arXiv2015

On the Green function of the annulus

Using the Gegenbauer polynomials and the zonal harmonics functions we give some representation formula of the Green function in the annulus. We apply this result to prove some uniqueness results for some nonlinear elliptic problems.

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