Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric
A. M. Micheletti Angela Pistoia
Discrete & Continuous Dynamical Systems - A 1998, 4(4): 709-720 doi: 10.3934/dcds.1998.4.709
Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$
keywords: transversality theorem. Riemannian metric multiple eigenvalues Laplace-Beltrami operator
Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents
Angela Pistoia Tonia Ricciardi
Discrete & Continuous Dynamical Systems - A 2017, 37(11): 5651-5692 doi: 10.3934/dcds.2017245

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 $γ∈(0, 1]$ 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 $\gamma=1$. Thus, by analyzing the case $\gamma≠1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena.

keywords: Asymmetric sinh-Poisson equation concentrating solution tower of bubbles
Supercritical problems in domains with thin toroidal holes
Seunghyeok Kim Angela Pistoia
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4671-4688 doi: 10.3934/dcds.2014.34.4671
In this paper we study the Lane-Emden-Fowler equation $$ (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. $$ Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $ (P)_ \epsilon$ increases as $\epsilon$ goes to zero.
keywords: concentration on $l-$dimensional manifolds. Supercritical problem
The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
Marco Ghimenti Anna Maria Micheletti Angela Pistoia
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2535-2560 doi: 10.3934/dcds.2014.34.2535
Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right. $$ and Schrödinger-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right. $$ when $p\in(2,6). $ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.
keywords: scalar curvature Lyapunov-Schmidt reduction. Klein-Gordon-Maxwell systems Scrhödinger-Maxwell systems Riemannian manifolds
The Morse property for functions of Kirchhoff-Routh path type
Thomas Bartsch Anna Maria Micheletti Angela Pistoia
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1867-1877 doi: 10.3934/dcdss.2019123
For a bounded domain
$ \Omega\subset\mathbb{R}^n $
$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $
be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary
$ {\mathcal C}^2 $
$ f:{\mathcal D}\to\mathbb{R} $
, defined on an open subset
$ {\mathcal D}\subset\mathbb{R}^{nN} $
, and fixed coefficients
$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $
we consider the function
$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $
defined as
$ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum\limits_{j,k = 1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). $
We prove that
$ f_\Omega $
is a Morse function for most domains
$ \Omega $
of class
$ {\mathcal C}^{m+2,\alpha} $
, any
$ m\ge0 $
$ 0<\alpha<1 $
. This applies in particular to the Robin function
$ h:\Omega\to\mathbb{R} $
$ h(x) = H_\Omega(x,x) $
, and to the Kirchhoff-Routh path function where
$ \Omega\subset\mathbb{R}^2 $
$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $
, and
$ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum\limits_{{j,k = 1}\atop{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|. $
keywords: Kirchhoff-Routh path function Morse function transversality theorem

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