American Institute of Mathematical Sciences

We perform a semiclassical analysis for the planar Schrödinger-Poisson system

\begin{document}$ \begin{gather} \begin{cases} -\varepsilon^{2} \Delta\psi+V(x)\psi = E(x) \psi \quad \text{in $ \mathbb{R}^2$}, \\ -\Delta E = |\psi|^{2} \quad \text{in $ \mathbb{R}^2$}, \end{cases} \end{gather}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S{P_\varepsilon }) $\end{document}

where \begin{document}$ \varepsilon $\end{document} is a positive parameter corresponding to the Planck constant and \begin{document}$ V $\end{document} is a bounded external potential. We detect solution pairs \begin{document}$ (u_\varepsilon, E_\varepsilon) $\end{document} of the system \begin{document}$ (SP_\varepsilon) $\end{document} as \begin{document}$ \ge \rightarrow 0 $\end{document}, leaning on a nongeneracy result in [3].

Given any \begin{document}$ \mu _1, \mu _2\in {\mathbb C} $\end{document} and \begin{document}$ \alpha >0 $\end{document}, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation \begin{document}$ \partial _{ tt } u - \Delta u + \mu _1 u = \mu _2 |u|^\alpha u $\end{document} on \begin{document}$ {\mathbb R}^N $\end{document}, \begin{document}$ N\ge 1 $\end{document}, that do not vanish, i.e. \begin{document}$ |u (t, x) | >0 $\end{document} for all \begin{document}$ x \in {\mathbb R}^N $\end{document} and all sufficiently small \begin{document}$ t $\end{document}. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

There are many natural phenomena that can best be described by the use of infinitesimal and infinite numbers (see e.g. [1,5,13,23]. However, until now, the Non-standard techniques have been applied to theoretical models. In this paper we investigate the possibility to implement such models in numerical simulations. First we define the field of Euclidean numbers which is a particular field of hyperreal numbers. Then, we introduce a set of families of Euclidean numbers, that we have called altogether algorithmic numbers, some of which are inspired by the IEEE 754 standard for floating point numbers. In particular, we suggest three formats which are relevant from the hardware implementation point of view: the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the Truncated Algorithmic Numbers. In the second part of the paper, we show a few applications of such numbers.

We give short survey on the question of asymptotic stability of ground states of nonlinear Schrödinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.

After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", as defined in the ETH approach. Our discussion is underpinned by a survey of results on causal fermion systems describing Minkowski space that show that an operator representing a spacetime point commutes with the algebra in its causal future, up to tiny corrections that depend on a regularization length.

We prove the existence of positive solutions for a class of semipositone problems with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.

Let \begin{document}$ (M,g) $\end{document} a compact Riemannian \begin{document}$ n $\end{document}-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of \begin{document}$ g $\end{document} there are scalar-flat metrics that have \begin{document}$ \partial M $\end{document} as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term \begin{document}$ h_{g} $\end{document} with a negative smooth function \begin{document}$ \alpha, $\end{document} the set of solutions of Yamabe problem is still a compact set.

We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:

\begin{document}$ \begin{equation*} \left\{ \begin{aligned} &\Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \ \ \ \text{in}\,\,\Omega,\\ &u = 0\quad\text{on}\,\,\partial\Omega, \end{aligned} \right. \end{equation*} $\end{document}

where \begin{document}$ \Omega\subset\mathbb{R}^N(N\geq3) $\end{document} is a bounded domain with smooth boundary, \begin{document}$ \lambda>0,\, r\in(2,4) $\end{document}. We prove as \begin{document}$ \lambda $\end{document} becomes large the existence of more and more sign-changing solutions of both positive and negative energies.