In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$

Computational methods and a program to obtain crystal structures that have the perfectly identical diffraction patterns, i.e. structure factors with the same absolute values and the same lattice symmetry are discussed. This is directly related to the uniqueness of solutions in crystal structure determination of single-crystal/powder-crystal samples from diffraction data. In order to solve the problem, it is necessary to solve a system of quadratic equations. The framework of positive-semidefinite programming is used herein to solve the system efficiently.

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

In this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

Let Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, . We obtain the existence of infinitely many solutions for the singular critical problem with Dirichlet boundary condition for suitable positive number λ.

In this work we deal with the numerical solution of aHamilton-Jacobi-Bellman (HJB) equation with infinitely manysolutions. To compute the maximal solution – the optimalcost of the original optimal control problem – we present acomplete discrete method based on the use of some finite elementsand penalization techniques.

The process of cataloguing the minor planet population of the Solar System has experienced a great advance in the last decades with the start-up of several surveys. The large volume of data generated by them has increased with time and given rise to huge databases of asteroids with uneven qualities.

In fact, a significant fraction of these objects have not been enough observed, thus leading to the computation of very poor quality orbits as to carry out useful predictions of the positions of such asteroids. As a result, some objects can get lost, which is particularly embarrassing for those with Earth crossing orbits.

When this situation persists for a long time, the aforementioned databases end up contaminated in the sense that they contain more than one discovery for the same physical object and some kind of action must be taken. The algorithms for asteroid identifications are thought precisely to mitigate this problem and their design will depend upon the quality of the available data for the objects to be identified.

In this paper we will distinguish two cases: when both objects have a nominal orbit and when one of them lacks it. In addition, when the available data poorly constrain the solution, other orbits in the neighbourhood of the nominal one are also compatible with the observations. Using these alternative orbits allows us to find many identifications that otherwise would be missed. Finally, we will show the efficiency of all these algorithms when applied to the datasets distributed by the Minor Planet Center.

Some of the problems related to Near Earth Objects (NEOs), like orbit determination and ephemeris computation, are not new, and had to be dealt with since the beginning of NEO astronomy. The latter practically started with the discovery of Comet D/1770 L1 Lexell, that passed very close to the Earth in 1770; studies of the chaotic dynamics of this exceptional object continued well into the XIXth century. At the end of the XXth century there has been a renewal of interest in NEOs, as attested by IAU Symposium 236.

When a celestial body, e.g., an asteroid, has been observed only over a short time, its orbit is not well determined but may be anywhere in a confidence region where the astrometric residuals are acceptable. This region can be sampled by a swarm of Virtual Asteroids (VA) sharing the reality of the asteroid: one of them is real, but we do not know which one. The problem is how to sample the confidence region with a small number of VA, still being able to solve the main problems of asteroid recovery/identification and impact monitoring.

One class of methods uses random sampling of the confidence region to mimic with the VA population the probabilistic distributions of the orbits. This class includes the Monte Carlo and the Statistical Ranging methods. When it is critical to detect a very small probability (e.g., of a catastrophic impact) by computing a small number of VA orbits, and also when a large catalog of asteroids has to be handled, it is more efficient to sample the confidence region with a geometric object, such as a smooth manifold: it can be sampled uniformly, taking into account its dimension. Our group has developed in the last 6-7 years 1-dimensional sampling methods based upon a differentiable curve, the Line Of Variations (LOV), which can represent, in suitable cases, the spine of the confidence region. The LOV is sampled by uniformly spaced VA, thus interpolation between consecutive VA is possible. This is the basis for the current algorithms of Impact Monitoring, used in Pisa and at JPL. The LOV method is also used for recovery of lost asteroids and for identification of independent discoveries of the same object.

When the asteroid has moved on the sky while being observed by $<1^\circ$, the confidence region is wide in two directions and the LOV may be an inappropriate way of sampling it. We have recently developed 2-dimensional sampling methods based upon the concept of Admissible Region, a 2-dimensional manifold parameterized by a compact subset of the range/range-rate plane. This region is then sampled by triangulation, with each node used as a VA. This allows to define methods for asteroid identification/recovery and for impact monitoring starting from very poor data, such as the ones collected during a single night of observations.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

In this paper we study the existence of multiple positive solutions and the bifurcation problem for the following equation:

$$ -\Delta u+u=\biggl(\int_{\mathbb{R}^3}\frac{|u(y)|^2}{|x-y|}\,\mathrm{d}y\biggr)u+\mu f(x),\quad x\in\mathbb{R}^3, $$

where $f(x)\in H^{-1}(\mathbb{R}^3)$, $f(x)\geq0$, $f(x)\not\equiv0$. We show that there are positive constants $\mu^{*}$ and $\mu^{**}$ such that the above equation possesses at least two positive solutions for $\mu\in(0,\mu^{*})$, and no positive solution for $\mu>\mu^{**}$. Furthermore, we prove that $\mu=\mu^{*}$ is a bifurcation point for the equation under study.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J70