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Filomena Pacella

Publications and source records attributed to Filomena Pacella.

At least 37 records · Page 2Linked to original sources
arXiv2019

New concentration phenomena for a class of radial fully nonlinear equations

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

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arXiv2019

Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

We study the critical Neumann problem \begin{equation*} \begin{cases} -Δu = |u|^{2^*-2}u &\text{in }Σ_ω,\\ \quad\frac{\partial u}{\partialν}=0 &\text{on }\partialΣ_ω, \end{cases} \end{equation*} in the unbounded cone $Σ_ω:=\{tx:x\inω\text{ and }t>0\}$, where $ω$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*:=\frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $ω$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $Σ_ω\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.

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arXiv2019

Isoperimetric cones and minimal solutions of partial overdetermined problems

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

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arXiv2019

Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in \cite{DaPaSys, DaGlPa1, Pa, PaWe}.

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arXiv2018

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain $Ω$ in a cone $Σ$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $Ω$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $Γ$ with boundary which satisfy a "gluing" condition with respect to the cone $Σ$. We prove that if either the cone is convex or the surface is a radial graph then $Γ$ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

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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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arXiv2016

Morse Index and Symmetry for Elliptic Problems with Nonlinear Mixed Boundary Conditions

Under a Morse index condition we prove symmetry results for solutions of a nonlinear mixed boundary condition elliptic problem. As an intermediate step we relate the Morse index of a solution to a mixed boundary condition linear eigenvalue problem for which we construct sequences of eigenvalues and provide variational characterization of them.

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arXiv2016

Existence results for fully nonlinear equations in radial domains

We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in $Ω$}\\ u=0 & \text{on $\partialΩ$} \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $Ω$ is either an annulus or a ball in $\Rn$, $n\geq2$. \\ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent $p>1$, if $Ω$ is an annulus; \item[ii)] existence of infinitely many sign changing radial solutions for every $p>1$, characterized by the number of nodal regions, if $Ω$ is an annulus; \item[iii)] existence of infinitely many sign changing radial solutions characterized by the number of nodal regions, if $F$ is one of the Pucci's operator, $Ω$ is a ball and $p$ is subcritical.

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arXiv2016

Asymptotic profile of positive solutions of Lane-Emden problems in dimension two

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= u^p & \mbox{ in }Ω\\ u>0 & \mbox{ in }Ω\\ u=0 & \mbox{ on }\partial Ω\end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded domain of $\mathbb R^2$. We give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty$, under the condition \[p\int_Ω |\nabla u_p|^2\,dx\rightarrow β\in\mathbb R\qquad\mbox{ as $p\rightarrow +\infty$}.\]

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arXiv2016

A Morse index formula for radial solutions of Lane-Emden problems

We consider the semilinear Lane-Emden problem: \begin{equation}\label{problemAbstract}\left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $1<p<p_S$, $p_S=\frac{N+2}{N-2}$. We prove that for any radial solution $u_p$ of \eqref{problemAbstract} with $m$ nodal domains its Morse index $\mathsf{m}(u_p)$ is given by the formula \[\mathsf{m}(u_p)=m+N(m-1)\] if $p$ is sufficiently close to $p_S$.

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arXiv2016

Asymptotic analysis for the Lane-Emden problem in dimension two

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }Ωu=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array}\right. \end{equation} when $p>1$ and $Ω\subset\mathbb R^2$ is a smooth bounded domain. The aim of the paper is to survey some recent results on the asymptotic behavior of solutions of (1) as the exponent $p\rightarrow \infty $.

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arXiv2016

Exact Morse index computation for nodal radial solutions of Lane-Emden problems

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\geq3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in $\mathbb R^N$ in any dimension $N\geq2$.

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arXiv2016

A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain

We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-Δu = | u |^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance: Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution.

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arXiv2015

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form $F(x,D^2u)+f(x,u)=0$, then the operator $\mathcal{L}_u=\mathcal{M}^++\frac{\partial f}{\partial u}(x,u)$, where $\mathcal{M}^+$ is the Pucci's sup--operator, plays the role of the linearized operator at $u$. In particular, we prove that if $u$ is a solution in a radial bounded domain, if $f$ is convex in $u$ and if the principal eigenvalue of $\mathcal{L}_u$ (associated with positive eigenfunctions) in any half domain is nonnegative, then $u$ is foliated Schwarz symmetric. We apply our symmetry results to obtain bounds on the spectrum and to deduce properties of possible nodal eigenfunctions for the operator $\mathcal{M}^+$.

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arXiv2015

Morse index of radial nodal solutions of Hénon type equations in dimension two

We consider non-autonomous semilinear elliptic equations of the type \[ -Δu = |x|^α f(u), \ \ x \in Ω, \ \ u=0 \quad \text{on} \ \ \partial Ω, \] where $Ω\subset {\mathbb R}^2$ is either a ball or an annulus centered at the origin, $α>0$ and $f: {\mathbb R}\ \rightarrow {\mathbb R}$ is $C^{1, β}$ on bounded sets of ${\mathbb R}$. We address the question of estimating the Morse index $m(u)$ of a sign changing radial solution $u$. We prove that $m(u) \geq 3$ for every $α>0$ and that $m(u)\geq α+ 3$ if $α$ is even. If $f$ is superlinear the previous estimates become $m(u) \geq n(u)+2$ and $m(u) \geq α+ n(u)+2$, respectively, where $n(u)$ denotes the number of nodal sets of $u$, i.e. of connected components of $\{ x\in Ω; u(x) \neq 0\}$. Consequently, every least energy nodal solution $u_α$ is not radially symmetric and $m(u_α) \rightarrow + \infty$ as $α\rightarrow + \infty$ along the sequence of even exponents $α$.

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arXiv2015

A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ Ω\\ u=0 & \hbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $λ>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $λ$ is close to zero there are no sign-changing solutions of the form $$u_λ=PU_{δ_1,ξ}-PU_{δ_2,ξ}+w_λ, $$ where $PU_{δ_i}$ is the projection on $H_0^1(Ω)$ of the regular positive solution of the critical problem in $\mathbb{R}^N$, centered at a point $ξ\in Ω$ and $w_λ$ is a remainder term. Some additional results on norm estimates of $w_λ$ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.

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