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### Open Access Journals

DCDS

We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\alpha < 0$, $\alpha \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set.
The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.

DCDS

We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with transmission trajectories, is continuous, though not differentiable, with respect to the initial data.

DCDS

This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper [1], where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.

DCDS

The field of nonlinear elliptic equations/systems has experienced a new burst of activities in recent years. This includes the resolution of De Giorgi's conjecture for Allen-Cahn equation,
the classification of stable/finite Morse index solutions for Lane-Emden equation, the regularity of interfaces of elliptic systems with large repelling parameter, Caffarelli-Silvestre extension of fractional laplace equations, the analysis of Toda type systems, etc.
This special volume touches several aspects of these new activities.

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DCDS

We characterize the set of harmonic functions with Dirichlet boundary conditions in unbounded
domains which are union of two different chambers. We analyse the asymptotic behavior of the solutions in connection with
the changes in the domain's geometry; we classify all (possibly sign-changing) infinite energy solutions having given asymptotic frequency at
the infinite ends of the domain; finally we sketch the case of several different chambers.

DCDS

Asymptotics of solutions to Schrödinger equations with
singular dipole-type potentials are investigated. We evaluate the
exact behavior near the singularity of solutions to elliptic
equations with potentials which are purely angular multiples of
radial inverse-square functions. Both the linear and the semilinear
(critical and subcritical) cases are considered.

DCDS

For a class of competition-diffusion nonlinear systems involving the $s$-power of the laplacian,
$s\in(0,1)$, of the form
\[
(-\Delta)^{s} u_i=f_i(u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k,
\]
we prove that $L^\infty$ boundedness implies $\mathcal{C}^{0,\alpha}$ boundedness for $\alpha>0$
sufficiently small, uniformly as $\beta\to +\infty$. This extends to the case $s\neq1/2$
part of the results obtained by the authors in the previous paper [arXiv: 1211.6087v1].

DCDS

The asymptotic behavior of solutions to Schrödinger
equations with singular homogeneous potentials is investigated.
Through an Almgren type monotonicity formula and separation of
variables, we describe the exact asymptotics near the singularity
of solutions to at most critical semilinear elliptic equations with
cylindrical and quantum multi-body singular
potentials. Furthermore, by an iterative Brezis-Kato procedure,
pointwise upper estimate are derived.

DCDS

In continuation of [20], we analyze the
properties of spectral minimal $k$-partitions of an open set
$\Omega$ in
$\mathbb R^3$
which are nodal, i.e. produced by the nodal domains of an
eigenfunction of the Dirichlet Laplacian in $\Omega$. We show that such a partition is necessarily a nodal partition
associated with a $k$-th eigenfunction. Hence we have in this case equality in Courant's nodal theorem.

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