arXiv subjects
Publications and source records attributed to Anna Lisa Amadori.
We investigate the (linearized) Morse index of solutions to Hamiltonan systems, with a focus on convex Hamiltonians functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions and give an estimate of their Morse index.
We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $α$. For sign-changing solutions, the case $α=0$ -- Lane-Emden equation -- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $p\mapsto u_p$, and the number of branching points increases with both the number of nodal zones and the exponent $α$. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between different branches, thus allowing to separate them at least in some cases.
We investigate nodal radial solutions to semilinear problems of type \[\begin{cases}-Δu = f(|x|,u) \qquad & \text{ in } Ω, \newline u= 0 & \text{ on } \partial Ω, \end{cases} \] where $Ω$ is a bounded radially symmetric domain of $\mathbb R^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work.
By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in a previous paper, we give a lower bound for the Morse index of radial solutions to Hénon type problems \[ \left\{\begin{array}{ll} -Δu = |x|^αf(u) \qquad & \text{ in } Ω, u= 0 & \text{ on } \partial Ω, \end{array} \right. \] where $Ω$ is a bounded radially symmetric domain of $\mathbb R^N$ ($N\ge 2$), $α>0$ and $f$ is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to $\infty$ as $α\to \infty$. Concerning the real Hénon problem, $f(u)= |u|^{p-1}u$, we prove radial nondegeneracy, we show that the radial Morse index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial.
In this paper we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent $p$ is close to $1$. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle $2π/n$. By considerations based on the Morse index we see that, depending on the values of $α$ and $n$, such least energy solutions can be radial, or nonradial and different one from another.
In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent p. As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.
We compute the Morse index of nodal radial solutions to the Hénon problem \[\left\{\begin{array}{ll} -Δu = |x|^α|u|^{p-1} u \qquad & \text{ in } B, \newline u= 0 & \text{ on } \partial B, \end{array} \right. \] where $B$ stands for the unit ball in ${\mathbb R}^N$ in dimension $N\ge 3$, $α>0$ and $p$ is near at the threshold exponent for existence of solutions $p_α=\frac{N+2+2α}{N-2}$, obtaining that \begin{align*} m(u_p) & = m \sum\limits_{j=0}^{1+\left[α/{2}\right]} N_j \quad & \mbox{ if $α$ is not an even integer, or} \newline m(u_p)& = m\sum\limits_{j=0}^{ α/2} N_j + (m-1) N_{1+α/ 2} & \mbox{ if $α$ is an even number.} \end{align*} Here $N_j$ denotes the multiplicity of the spherical harmonics of order $j$. The computation builds on a characterization of the Morse index by means of a one dimensional singular eigenvalue problem, and is carried out by a detailed picture of the asymptotic behavior of both the solution and the singular eigenvalues and eigenfunctions. In particular it is shown that nodal radial solutions have multiple blow-up at the origin, where each node converges (up to a suitable rescaling) to the bubble shaped solution of a limit problem. As side outcome we see that solutions are nondegenerate for $p$ near at $p_α$, and we give an existence result in perturbed balls.
We propose a stochastic model in evolutionary game theory where individuals (or subpopulations) can mutate changing their strategies randomly (but rarely) and explore the external environment. This environment affects the selective pressure by modifying the payoff arising from the interactions between strategies. We derive a Fokker-Plank integro-differential equation and provide Monte-Carlo simulations for the Hawks vs Doves game. In particular we show that, in some cases, taking into account the external environment favors the persistence of the low-fitness strategy.
In this paper we prove the existence of continua of nonradial solutions for the Lane-Emden equation. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane Emden problem has a very rich and complex structure.
In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -Δu = |u|^{p-1} u \qquad & \text{in} Ω, \\ u= 0 & \text{on} \partialΩ, \end{array} \right. $$ where $Ω$ is a bounded domain in ${\mathbb R}^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1 1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as $p\to 1$. Keywords: semilinear elliptic equations, nodal solutions, supercritical problems.
In this paper we consider the problem $-Δu=|x|^α F(u)$ in $R^N$, with $α>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $α$ is an even integer.
In this paper we consider the Hénon problem in a ball. We prove the existence of (at least) one branch of nonradial solutions that bifurcate from the radial ones and that this branch is unbounded.
In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.