arXiv subjects
Publications and source records attributed to Angela Pistoia.
For each $n\geq 3$ we establish the existence of a nodal solution $u$ to the Yamabe problem on the round sphere $(\mathbb{S}^n,g)$ which satisfies $$\int_{\mathbb{S}^n}|u|^{2^*}dV_g < 2m_n\mathrm{vol}(\mathbb{S}^n),$$ where $m_3=9,$ $m_4= 7,$ $m_5=m_6=6$, and $m_n= 5\ \text{if }n\geq 7.$
We prove the existence of solutions $(λ, v)\in \mathbb{R}\times H^{1}(Ω)$ of the elliptic problem \[ \begin{cases} -Δv+(V(x)+λ) v =v^{p}\ &\text{ in $ Ω, $} \ v>0,\qquad \int_Ωv^2\,dx =ρ. \end{cases} \] Any $v$ solving such problem (for some $λ$) is called a normalized solution, where the normalization is settled in $L^2(Ω)$. Here $Ω$ is either the whole space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions. Moreover, $1 1$ if $N=1,2$. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $Ω$ as the prescribed mass $ρ$ is either small (when $p<1+\frac 4N$) or large (when $p>1+\frac 4N$) or it approaches some critical threshold (when $p=1+\frac 4N$).
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.
We prove the non-degeneracy for the critical Lane--Emden system $$ -ΔU = V^p,\quad -ΔV = U^q,\quad U, V > 0 \quad \text{in } \mathbb{R}^N $$ for all $N \ge 3$ and $p,q > 0$ such that $\frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}$. We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane-Emden system provided that they belong to the corresponding energy space or they decay to 0 uniformly as the point tends to infinity.
We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -Δu=λ_1\dfrac{V_1 e^{u}}{ \int_{Ω_{\boldsymbolε}} V_1 e^{u} dx } - λ_2τ\dfrac{ V_2 e^{-τu}}{ \int_{Ω_{\boldsymbolε}}V_2 e^{ - τu} dx}&\text{in $Ω_{\boldsymbolε}=Ω\setminus \displaystyle \bigcup_{i=1}^m \overline{B(ξ_i,ε_i)}$}\\ \ \ u=0 &\text{on $\partial Ω_{\boldsymbolε}$}, \end{array} \right. $$ where $B(ξ_i,ε_i)$ is a ball centered at $ξ_i\inΩ$ with radius $ε_i$, $τ$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $λ_1>8πm_1$ and $λ_2 τ^2>8π(m-m_1)$ with $m_1 \in \{0,1,\dots,m\}$, there exist radii $ε_1,\dots,ε_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $ξ_1,\dots,ξ_{m_1}$ and $ξ_{m_1+1},\dots,ξ_{m}$, respectively, as the radii approach zero.
We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -Δu = v^p &\text{in } \mathbb{R}^n,\\ -Δv = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] where $n \ge 3$ and $p,q>0$ and $(p,q)$ belongs to the critical hyperbola $\frac{1}{p+1} + \frac{1}{q+1} = \frac{n-2}{n}.$ We prove that they are non-degenerate when either $(p,q)$ is close to $(1,{n+4\over n-4})$ (if $n\ge5$) or $(p,q)$ is close to $({n+2\over n-2},{n+2\over n-2})$ (if $n\ge3$).
We consider the nonlinear Schrödinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e. an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the real line. On the other hand, a Ljapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges.
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.
We consider the following critical elliptic system: \begin{equation*} \begin{cases} -Δu_i=μ_i u_i^{3}+βu_i^{ } \sum\limits_{j\neq i} u_j^{2} \quad \hbox{in}\ Ω_\varepsilon \\ u_i=0 \hbox{ on } \partialΩ_\varepsilon , \qquad u_i>0 \hbox{ in } Ω_\varepsilon \end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain $Ω_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole $B_\varepsilon(ξ_0)$. For $μ_i>0$, $β<0$, and $\varepsilon>0$ small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component $u_i$ exhibits a towering blow-up around $ξ_0$ as $\varepsilon \to 0$. The proof is based on the Ljapunov-Schmidt reduction method, and the velocity of concentration of each layer within a given tower is chosen in such a way that the interaction between bubbles of different components balance the interaction of the first bubble of each component with the boundary of the domain, and in addition is dominant when compared with the interaction of two consecutive bubbles of the same component.
We consider equations of the form $Δu +λ^2 V(x)e^{\,u}=ρ$ in various two dimensional settings. We assume that $V>0$ is a given function, $λ>0$ is a small parameter and $ρ=\mathcal O(1)$ or $ρ\to +\infty$ as $λ\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $ρ=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.
In this paper we consider the Liouville equation $Δu +λ^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $Ω$. We show that there exists a simple, closed curve $γ\subset Ω$ such that for a sequence $λ_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{λ_n}}\to H$, where $H$ is a harmonic function in $Ω\setminusγ$ and $\frac{λ_n^2}{\log\frac{1}{λ_n}}\int_Ωe^{\,u_n}\,dx\to 8πc_Ω$, where $c_Ω$ is a constant depending on the conformal class of $Ω$ only.
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10.
In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -Δu&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }Ω,\\ u&=0, &\hbox{ on }\partialΩ. \end{array}\right. \eqno(P) \end{equation*} on the annulus $Ω:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $N\geq 3.$ In particular, for any integer $k$ large enough, we build a non-radial solution which look like the unique positive solution $u_0$ to $(P)$ crowned by $k$ negative bubbles arranged on a regular polygon with radius $r_0$ such that $r_0^{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\max_{a\leq r\leq b}r^{\frac{N-2}{2}}u_0(r).$
We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-Δu=μ_{1}|u|^{{2}^{\ast}-2}u+λα|u|^{α-2}|v|^{β}u,$\\ $-Δv=μ_{2}|v|^{{2}^{\ast}-2}v+λβ|u|^α|v|^{β{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. \] where $N\geq4,$ $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $α,β\in(1,2],$ $α+β=2^{\ast},$ $μ_{1},μ_{2}>0,$ and $λ<0.$ We show that these solutions exhibit phase separation as $λ\rightarrow-\infty,$ and we give a precise description of their limit domains. If $μ_{1}=μ_{2}$ and $α=β$, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.
We consider the following perturbed critical Dirichlet problem involving the Hardy-Schrödinger operator on a smooth bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 3$, with $0 \in Ω$: $$ \left\{ \begin{array}{ll}-Δu-γ\frac{u}{|x|^2}-εu=|u|^{\frac{4}{N-2}}u &\hbox{in }Ωu=0 & \hbox{on }\partial Ω, \end{array}\right. $$ when $ε>0$ is small and $γ< {(N-2)^2\over4}$. Setting $ γ_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $γ\leq \frac{(N-2)^2}{4}-1$ and $γ\neq γ_j$ for any $j$, then for small $ε$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $γ<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $ε$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $γ\neq γ_j$ is not necessary. Indeed, it is known that, if $γ> \frac{(N-2)^2}{4}-1$ and $Ω$ is a ball $B$, then there is no radial positive solution for $ε>0$ small. We complete the picture here by showing that, if $γ\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $ε>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $γ=0$.
For a bounded domain $Ω\subset\mathbb{R}^n$ let $H_Ω:Ω\timesΩ\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $λ_1,\dots,λ_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_Ω:{\mathcal D}\capΩ^N\to\mathbb{R}$ defined as \[ f_Ω(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N λ_jλ_k H_Ω(x_j,x_k). \] We prove that $f_Ω$ is a Morse function for most domains $Ω$ of class ${\mathcal C}^{m+2,α}$, any $m\ge0$, $0<α<1$. This applies in particular to the Robin function $h:Ω\to\mathbb{R}$, $h(x)=H_Ω(x,x)$, and to the Kirchhoff-Routh path function where $Ω\subset\mathbb{R}^2$, ${\mathcal D}=\{x\in\mathbb{R}^{2N}: \text{$x_j\ne x_k$ for $j\ne k$}\}$, and \[ f(x_1,\dots,x_N) = - \frac{1}{2π}\sum_{\genfrac{}{}{0pt}{}{j,k=1}{j\ne k}}^Nλ_jλ_k\log|x_j-x_k|. \]
Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a smooth function on $M$ and assume that it has a critical point $ξ\in M$ such that $h(ξ)=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_λ=u_λ^\frac4{n-2}g$, with $u>0$, whose scalar curvature is the prescribed function $h_λ=λ^2+h$, where $λ$ is a small parameter. In the positive case, i.e. when the scalar curvature $R_g$ is strictly positive, we find a family of bubbling metrics $g_λ$, where $u_λ$ blows-up at the point $ξ$ and approaches zero far from $ξ$ as $λ$ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric $g_0=u_0^\frac4{n-2}g$, with $u_0>0$, whose scalar curvature is equal to $h$, then there exists a bounded family of conformal metrics $g_{0,λ}=u_{0,λ}^\frac4{n-2}g$, with $u_{0,λ}>0$, which satisfies $u_{0,λ}\to u_0$ uniformly as $λ\to 0$. Here, we build a second family of bubbling metrics $g_λ$, where $u_λ$ blows-up at the point $ξ$ and approaches $u_0$ far from $ξ$ as $λ$ goes to zero. In particular, this shows that this problem admits more than one solution.
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.