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Angela Pistoia

Publications and source records attributed to Angela Pistoia.

At least 55 records · Page 3Linked to original sources
arXiv2022

Phase separating solutions for two component systems in general planar domains

In this paper we consider a two component system of coupled non linear Schrödinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions in the limit of large interspecies scattering length $β$ the system reduces to a couple of scalar problems on subdomains of pure phases. Here we show that given a solution to the limiting problem under some additional non degeneracy assumptions there exists a family of solutions parametrized by the parameter $β\gg 1$.

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arXiv2022

Clustering phenomena in low dimensions for a boundary Yamabe problem

We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.

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arXiv2022

Segregated solutions for a critical elliptic system with a small interspecies repulsive force

We consider the elliptic system $$-Δu_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ β_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $α:=β_{ij}$ and $β:=β_{i(q+1)}=β_{(q+1)j}$ for any $i,j=1,\dots,q.$ If $β<0$ and $|β|$ is small enough we build solutions such that each component $u_{1},\dots,u_q$ blows-up at the vertices of $q$ polygons placed in different great circles which are linked to each other, and the last component $u_{q+1}$ looks like the radial positive solution of the single equation.

math.AP↗
arXiv2021

A blow-up phenomenon for a non-local Liouville-type equation

We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature function, provided some generic assumptions are satisfied.

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arXiv2021

Yamabe systems and optimal partitions on manifolds with symmetries

We prove the existence of regular optimal $G$-invariant partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$ when $G$ is a compact group of isometries of $M$ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy $G$-invariant solution with nontrivial components and we show that the limit profiles of the its components separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing $G$-invariant solution to the Yamabe equation with precisely two nodal domains.

math.AP↗
arXiv2021

Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system $$-Δu_i+ λ_iu_i = \sum\limits_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ Ω, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $Ω$ in $\mathbb R^N,$ $N\ge 3$. The $λ_i$'s are real numbers, and the nonlinear term may have subcritical ($1 \frac{N}{N-2}$). The matrix $(β_{ij})$ is symmetric and admits a block decomposition such that the diagonal entries $β_{ii}$ are positive, the interaction forces within each block are attractive (i.e., all entries $β_{ij}$ in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component $u_i$ is nontrivial. We also find fully synchronized solutions (i.e., $u_i=c_i u_1$ for all $i=2,\ldots,\ell$) in the purely cooperative case whenever $p\in(1,2).$

math.AP↗
arXiv2021

Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

math.AP↗
arXiv2021

Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

We study the existence of solutions to the cubic Schrödinger system $$ -Δu_i = \sum_{j =1}^m β_{ij} u_j^2u_i + λ_i u_i\ \hbox{in}\ Ω,\ u_i=0\ \hbox{on}\ \partialΩ,\ i =1,\dots,m, $$ when $Ω$ is a bounded domain in $\mathbb R^4, $ $λ_i$ are positive small numbers, $β_{ij}$ are real numbers so that $β_{ii}>0$ and $β_{ij}=β_{ji}$, $i\neq j$. We assemble the components $u_i$ in groups so that all the interaction forces $β_{ij}$ among components of the same group are attractive, i.e. $β_{ij}>0$, while forces among components of different groups are repulsive or weakly attractive, i.e. $β_{ij}<\overlineβ$ for some $\overlineβ$ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters $λ_i$'s approach zero.

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arXiv2020

Large conformal metrics with prescribed Gaussian and geodesic curvatures

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-Δu=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_νu+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.

math.AP↗
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].

math.AP↗
arXiv2020

Nodal solutions of the Brezis-Nirenberg problem in dimension 6

We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$

math.AP↗
arXiv2020

A solution to a slightly subcritical elliptic problem with non-power nonlinearity

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.

math.AP↗
arXiv2020

Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

math.AP↗
arXiv2020

Singular Yamabe metrics by equivariant reduction

We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients.

math.AP↗
arXiv2020

Multiple blowing-up solutions to critical elliptic systems in bounded domains

We construct families of blowing-up solutions to elliptic systems on smooth bounded domains in the Euclidean space, which are variants of the critical Lane-Emden system and analogous to the Brezis-Nirenberg problem. We find a function which governs blowing-up points and rates, observing that it reflects the strong nonlinear characteristic of the system. By using it, we also prove that a single blowing-up solution exists in general domains, and construct examples of contractible domains where multiple blowing-up solutions are allowed to exist. We believe that a variety of new ideas and arguments developed here will help to analyze blowing-up phenomena in related Hamiltonian-type systems.

math.AP↗
arXiv2019

Nondegeneracy of the bubble for the critical p-Laplace equation

We prove the non-degeneracy of the extremals of the Sobolev inequality $$\int\limits_{\mathbb R^N}|\nabla u|^pdx\ge \mathcal S_p\int\limits_{\mathbb R^N}|u|^{Np\over N-p}dx,\ u\in \mathcal D^{1,p}(\mathbb R^N)$$ when $1<p<N,$ as solutions of a critical quasilinear equation involving the $p-$Laplacian.

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