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Angela Pistoia

Publications and source records attributed to Angela Pistoia.

At least 91 records · Page 5Linked to original sources
arXiv2016

On Coron's problem for weakly coupled elliptic systems

We consider the following critical weakly coupled elliptic system \[ \begin{cases} -Δu_i = μ_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} β_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in $Ω_\varepsilon$} u_i >0 & \text{in $Ω_\varepsilon$} u_i = 0 & \text{on $\partial Ω_\varepsilon$},\end{cases} \qquad i =1,\dots,m, \] in a domain $Ω_\varepsilon \subset \mathbb{R}^N$, $N=3,4$, with small shrinking holes as the parameter $\varepsilon \to 0$. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

math.AP↗
arXiv2016

A unified approach of blow-up phenomena for two-dimensional singular Liouville systems

We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.

math.AP↗
arXiv2016

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $γ\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $γ=1$. Thus, by analyzing the case $γ\neq1$ we emphasize specific properties of the physically relevant parameter $γ$ in the vortex concentration phenomena.

math.AP↗
arXiv2016

Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

In this paper we deal with the nonlinear Schrödinger system \[ -Δu_i =μ_i u_i^3 + βu_i \sum_{j\neq i} u_j^2 + λ_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(Ω) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($β<0$ fixed or $β\to -\infty$) or in the weakly cooperative case ($β\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $Ω$, families of positive solutions which blowup and concentrate at different points as $λ_1,\ldots, λ_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.

math.AP↗
arXiv2016

New blow-up phenomena for SU(n+1) Toda system

We consider the $SU(n+1)$ Toda system $$(S_λ) \quad \left\{ \begin{aligned} & Δu_1 + 2λe^{u_1} - λe^{u_2}- \dots - λe^{u_k} = 0\quad \hbox{in}\ Ω,\\ & Δu_2 - λe^{u_1} + 2λe^{u_2} - \dots - λe^{u_k}=0\quad \hbox{in}\ Ω,\\ &\vdots \hskip3truecm \ddots \hskip2truecm \vdots\\ & Δu_k -λe^{u_1}-λe^{u_2}- \dots+2λe^{u_k}=0\quad \hbox{in}\ Ω, &u_1 = u_2 = \dots = u_k =0 \quad \hbox{on}\ \partialΩ.\\ \end{aligned}\right. $$ If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, we construct a family of solutions $({u_1}_λ,\dots,{u_k}_λ)$ to $(S_λ)$ such that the $i-$th component ${u_i}_λ$ blows-up at the origin with a mass $2^{i+1}π$ as $λ$ goes to zero.

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arXiv2016

Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem $$(P)\qquad - Δv + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_νv = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D} $ is an open bounded domain in $\mathbb{R}^N,$ $ν$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal D,$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial\mathcal D$ as $q$ approaches from either below or above the higher critical Sobolev exponent ${2(N-h)\over N-h-2}.$

math.AP↗
arXiv2016

Boundary concentration phenomena for the higher-dimensional Keller-Segel system

We study the existence of steady states to the Keller-Segel system with linear chemotactical sensitivity function on a smooth bounded domain in $\mathbb R^N,$ $N\ge3,$ having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable $(N-2)-$dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.

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arXiv2016

Large mass boundary condensation patterns in the stationary Keller-Segel system

We consider the boundary value problem $-Δu + u =λe^u$ in $Ω$ with Neumann boundary condition, where $Ω$ is a bounded smooth domain in $\mathbb R^2$, $λ>0.$ This problem is equivalent to the stationary Keller-Segel system from chemotaxis. We establish the existence of a solution $u_λ$ which exhibits a sharp boundary layer along the entire boundary $\partialΩ$ as $λ\to 0$. These solutions have large mass in the sense that $ \int_Ωλe^{u_λ} \sim |\logλ|.$

math.AP↗
arXiv2016

Towering phenomena for the Yamabe equation on symmetric manifolds

Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $ξ_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_ε)\qquad-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $ε_k>0$ such that for all $ε\in (0, ε_k)$ the problem $(P_ε)$ has a symmetric solution $u_ε,$ which looks like the superposition of $k$ positive bubbles centered at the point $ξ_0$ as $ε\to 0$. In particular, $ξ_0$ is a {\em towering} blow-up point.

math.AP↗
arXiv2015

Clustering phenomena for linear perturbation of the Yamabe equation

Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $$-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) $$ where the first eigenvalue of the conformal laplacian $-\mathcal L_g $ is positive and $ε$ is a small positive parameter. We prove that for any point $ξ_0\in M$ which is non-degenerate and non-vanishing minimum point of the Weyl's tensor and for any integer $k$ there exists a family of solutions developing $k$ peaks collapsing at $ξ_0$ as $ε$ goes to zero. In particular, $ξ_0$ is a non-isolated blow-up point.

math.AP↗
arXiv2015

Concentration along geodescis for a nonlinear Steklov problem arising in corrosion modelling

We consider the problem of finding pairs $(λ; u)$, with $λ> 0$ and $u$ a harmonic function in a three dimensional torus-like domain, satisfying the nonlinear boundary condition $\partial_ν u = λ\sinh u$ on the boundary. This type of boundary condition arises in corrosion modelling (Butler Volmer condition). We prove existence of solutions which concentrate along some geodesics of the boundary as the parameter $λ$ goes to zero.

math.AP↗
arXiv2015

On Yamabe type problems on Riemannian manifolds with boundary

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -Δ_{g}u+au=0 & \text{ on }M \\ \partial_νu+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $ν$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.

math.AP↗
arXiv2015

Symmetries, Hopf fibrations and supercritical elliptic problems

We consider the semilinear elliptic boundary value problem \[ -Δu=\left\vert u\right\vert ^{p-2}u\text{ in }Ω,\text{\quad }u=0\text{ on }\partialΩ, \] in a bounded smooth domain $Ω$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

math.AP↗
arXiv2015

Ground states of critical and supercritical problems of Brezis-Nirenberg type

We study the existence of symmetric ground states to the supercritical problem \[ -Δv=λv+\left\vert v\right\vert ^{p-2}v\text{ \ in }Ω,\qquad v=0\text{ on }\partialΩ, \] in a domain of the form \[ Ω=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \inΘ\}, \] where $Θ$ is a bounded smooth domain such that $\overlineΘ \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $λ\in\mathbb{R},$ and $p=\frac{2(N-k)}{N-k-2}$ is the $(k+1)$-st critical exponent. We show that symmetric ground states exist for $λ$ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval $(-\infty,λ_{\ast})$ with $λ_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem \[ -\text{div}(a(x)\nabla u)=λb(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ Θ,\qquad u=0\quad\text{on}\ \partialΘ, \] where $a,b,c$ are positive continuous functions on $\overlineΘ.$ We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of $λ,$ and obtain a bifurcation result for ground states.

math.AP↗
arXiv2014

Asymmetric blow-up for the SU(3) Toda System

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

math.AP↗
arXiv2014

Concentration on minimal submanifolds for a Yamabe type problem

We construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.

math.AP↗
arXiv2014

Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_Ω(x_1,\ldots, x_N) =\sum\limits^N_{i=1}Γ^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N Γ_iΓ_jG(x_i,x_j)+2\sum\limits_{i=1}^NΓ_iψ_0(x_i)$ in a bounded domain $Ω\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $Ω$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $ψ_0\in C^1(\overlineΩ)$ is a harmonic function on $Ω$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under conditions on the vorticities $Γ_i\in{\mathbb R}\setminus\{0\}$. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case $Γ_i=(-1)^i$ of counter-rotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.

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