arXiv subjects

Alberto Boscaggin

Publications and source records attributed to Alberto Boscaggin.

43 records · Page 3Linked to original sources

arXiv2016

Scattering parabolic solutions for the spatial N-centre problem

For the $N$-centre problem in the three dimensional space, $$ \ddot x = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{α+2}}, \qquad x \in \mathbb{R}^3 \setminus \{c_1,\ldots,c_N\}, $$ where $N \geq 2$, $m_i > 0$ and $α\in [1,2)$, we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.

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arXiv2015

Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( λa^{+}(t) - μa^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $λ, μ$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

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arXiv2015

Positive solutions to indefinite Neumann problems when the weight has positive average

We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$ u" + q(t)g(u) = 0, \quad t \in [0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$ a continuous function defined on the whole real line.

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arXiv2015

Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + λa(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $λ> 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

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arXiv2015

Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation

We deal with the singularly perturbed Nagumo-type equation $$ ε^2 u'' + u(1-u)(u-a(s)) = 0, $$ where $ε> 0$ is a real parameter and $a: \mathbb{R} \to \mathbb{R}$ is a piecewise constant function satisfying $0 < a(s) < 1$ for all $s$. We prove the existence of chaotic, homoclinic and heteroclinic solutions, when $ε$ is small enough. We use a dynamical systems approach, based on the Stretching Along Paths method and on the Conley-Wazewski's method.

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arXiv2014

Highly oscillatory solutions of a Neumann problem for a $p$-laplacian equation

We deal with a boundary value problem of the form $-ε(ϕ_p(εu'))'+a(x)W'(u)=0,\quad u'(0)=0=u'(1),$ where $ϕ_p(s) = \vert s \vert^{p-2} s$ for $s \in \mathbb{R}$ and $p>1$, and $W:[-1,1] \to {\mathbb R}$ is a double-well potential. We study the limit profile of solutions when $ε\to 0^+$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $ε$ is small enough.

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arXiv2014

Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - μa^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $μ>0$ is large. Such solutions are characterized by the fact of being either small or large in each interval of positivity of $a$. In this way, we find periodic solutions, having minimal period arbitrarily large, and bounded non-periodic solutions, exhibiting a complex behavior. The proof is variational, exploiting suitable natural constraints of Nehari type.

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