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Angela Pistoia

Publications and source records attributed to Angela Pistoia.

At least 109 records · Page 6Linked to original sources
arXiv2014

Bubbling solutions for supercritical problems on manifolds

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $Γ$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-Δ_gu+h u=u^{\frac{n+1}{n-3}\pmε},\ u>0,\ \hbox{in}\ (M,g)$$ has a solution that concentrates along $Γ$ as $ε$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ satisfy a suitable condition. A connection with the solution of a class of periodic O.D.E.'s with singularity of attractive or repulsive type is established.

math.AP↗
arXiv2014

Blowing-up solutions concentrating along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let $(M,g)$ and $(K,κ)$ be two Riemannian manifolds of dimensions $m$ and $k ,$ respectively. Let $ω\in C^2(N),$ $ω> 0.$ The warped product $ M\times _ωK$ is the $ (m+k)$-dimensional product manifold $M\times K$ furnished with metric $ g+ω^2 κ.$ We prove that the supercritical problem $$-Δ_{g+ω^2 κ}u+h u=u^{ {m+2\over m-2} \pm\varepsilon},\ u>0,\ \hbox{in}\ (M\times _ωK,g+ω^2 κ)$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $Γ$ of $M\times _ωN$ as the real parameter $\varepsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ satisfy a suitable condition.

math.AP↗
arXiv2013

On the supercritical mean field equation on pierced domains

We consider the problem $$(P_\eps)\qquad Δu +λ{e^{u}\over\int\limits_{Ω\setminus B(ξ,\eps)}e^u}=0\ \hbox{in}\ Ω\setminus B(ξ,\eps),\quad u =0\ \hbox{on}\ \partial\(Ω\setminus B(ξ,\eps)\), $$ where $Ω$ is a smooth bounded open domain in $\rr^2$ which contains the point $ξ.$ We prove that if $λ>8π,$ problem $(P_\eps)$ has a solutions $u_\eps$ such that $$u_\eps(x)\to {8π+ λ\over2} G(x,ξ) \ \hbox{uniformly on compact sets of $Ω\setminus\{ξ\}$}$$ as $\eps$ goes to zero. Here $G$ denotes Green's function of Dirichlet Laplacian in $Ω.$ If $λ\not\in 8π\mathbb N$ we will not make any symmetry assumptions on $Ω,$ while if $λ\in 8π\mathbb N$ we will assume that $Ω$ is invariant under a rotation through an angle ${8π^2\over \la} $ around the point $ξ.$

math.AP↗
arXiv2013

Supercritical problems on manifolds

Let $(M,g)$ be a $m$-dimensional compact Riemannian manifold without boundary. Assume $κ\in C^2(M)$ is such that $-Δ_g+κ$ is coercive. We prove the existence of a solution to the supercritical problems $$ -Δ_gu+κu= u^p,\ u>0\quad\hbox{in}\ (M,g)\quad\hbox{and}\quad-Δ_gu+κu=λe^u\quad\hbox{in}\ (M,g) $$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\to\infty$ and $λ\to0$, respectively, under suitable symmetry assumptions on the manifold $M$.

math.AP↗
arXiv2013

Supercritical problems in domains with thin toroidal holes

In this paper we study the Lane-Emden-Fowler equation $$(P)_ε \{Δu+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_ε, u=0 \ \hbox{on}\ \partial\mathcal D_ε.$$ Here $\mathcal D_ε= \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,Γ_\ell)\le ε\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $Γ_\ell$ is an $\ell-$dimensional closed manifold such that $Γ_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_ε$ increases as $ε$ goes to zero.

math.AP↗
arXiv2013

Boundary towers of layers for some supercritical problems

We show that in some suitable torus-like domains D some supercritical elliptic problems have an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a k-dimensional submanifold of the boundary of D as p approaches 2*_{N,K} from below.

math.AP↗
arXiv2013

Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole

We consider the supercritical problem \[ -Δv=|v|^{p-2}v in Θ_ε, v=0 on \partialΘ_ε, \] where $Θ$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$, and $Θ_ε$ is obtained by deleting the $ε$-neighborhood of some sphere which is embedded in $Θ$. In some particular situations we show that, for $ε>0$ small enough, this problem has a positive solution $v_ε$ and that these solutions concentrate and blow up along the sphere as $ε$ tends to 0. Our approach is to reduce this problem to a critical problem of the form \[ -Δu=Q(x)|u|^{4/(n-2)}u in Ω_ε, u=0 on \partialΩ_ε, \] in a punctured domain $Ω_ε:=\{x\inΩ:|x-ξ_{0}|>ε\}$ of lower dimension, by means of some Hopf map. We show that, if $Ω$ is a bounded smooth domain in $\mathbb{R}^{n}$, $n\geq3$, $ξ_{0} is inΩ,$ $Q is in C^{2}(\b{\Oarmega})$ is positive and $\nabla Q(ξ_{0})\neq0$ then, for $ε>0$ small enough, this problem has a positive solution $u_ε$, and that these solutions concentrate and blow up at $ξ_{0}$ as $ε$ goes to 0.

math.AP↗
arXiv2013

The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of singularly perturbed Klein-Gordon-Maxwell systems and Schroedinger-Maxwell systems on M, with a subcritical nonlinearity. We prove that when the perturbation parameter epsilon is small enough, any stable critical point x_0 of the scalar curvature of the manifold (M,g) generates a positive solution (u_eps,v_eps) to both the systems such that u_eps concentrates at xi_0 as epsilon goes to zero.

math-ph↗
arXiv2013

Bubble concentration on spheres for supercritical elliptic problems

We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -Δv= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and $p_\eps={(m+1)+2\over(m+1)-2}-\eps$, $\eps>0.$ We prove the existence of positive and sign changing solutions of $(P_\eps)$ concentrating and blowing-up, as $\eps\to0$, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem $(P_\eps)$ into a nonhomogeneous problem in an annulus $\mathcal D\subset \rr^{m+1}$ which can be solved by a Ljapunov-Schmidt finite dimensional reduction.

math.AP↗
arXiv2012

Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -Δu = |u|^{p-2}u in Ω, u=0 on \partialΩ, where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $Ω$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.

math.AP↗
arXiv2012

Boundary clustered layers near the higher critical exponents

We consider the supercritical problem {equation*} -Δu=|u| ^{p-2}u\text{\in}Ω,\quad u=0\text{\on}\partialΩ, {equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\ast}:=\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\mathbb{R}^{N-k})$ in $L^{q}(\mathbb{R}^{N-k})$, $1\leq k\leq N-3.$ We show that in some suitable domains $Ω$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\partialΩ$ as $p$ approaches $2_{N,k}^{\ast}$ from below. Key words:Nonlinear elliptic boundary value problem; critical and supercritical exponents; existence of positive and sign changing solutions.

math.AP↗
arXiv2012

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$Δ_g u+(N-2/4(N-1) S_g+εh)u=u^{N+2/N-2} in M, u>0 in M,$$ where Δ_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,α}(M)$, and $ε$ is a small parameter.

math.AP↗
arXiv2012

The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). We show that a-priori L^\infty-bounds fail for linear perturbations on all manifolds with n\geq 4 as well as a-priori gradient L^2--bounds fail for non-locally conformally flat manifolds with n\geq 6 and for locally conformally flat manifolds with n\geq 7. In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g.

math.AP↗
arXiv2012

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem {equation*}\e^2Δv-v+f(v)=0\hbox{in}Ω,\quad v=0 \hbox{on}\partial Ω,{equation*} where $Ω$ is a smooth and bounded domain of $\R^N$, $\e$ is a small positive parameter, $f$ is a superlinear, subcritical and odd nonlinearity. In particular we prove that if $Ω$ has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that $k$ is even and sufficiently large, a $k$-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of $\partial Ω$.

math.AP↗
arXiv2012

Multiple blow-up phenomena for the sinh-Poisson equation

We consider the sinh-Poisson equation $$(P)_λ\quad -Δu=\la\sinh u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\rr^2$ and $λ$ is a small positive parameter. If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, for any integer $k$ if $\la$ is small enough, we construct a family of solutions to $(P)_\la$ which blows-up at the origin whose positive mass is $4πk(k-1)$ and negative mass is $4πk(k+1).$ It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].

math.AP↗