A mass supercritical and Sobolev critical fractional Schrödinger system

We study the following coupled fractional Schrödinger system: $$ \bcs (-\De)^s u=\la_1 u+\mu_1|u|^{p-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2}\quad &\hbox{in}\;\mathbb{R}^N, \\ (-\De)^s v=\la_2 v+\mu_2|v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &\hbox{in}\;\mathbb{R}^N, \\ %\int_{\mathbb{R}^N} u^2=a\quad and\quad \int_{\mathbb{R}^N} v^2=b, \ecs $$ with prescribed mass \[ \int_{\mathbb{R}^N} u^2=a\quad \hbox{and}\quad \int_{\mathbb{R}^N} v^2=b. \] Here, $a, b>0$ are prescribed, $N>2s, s>\frac{1}{2}$, $2+\frac{4s}{N}0$ sufficiently large, a mountain pass-type normalized solution exists provided $2\leq N\leq 4s$ and $ 2+\frac{4s}{N}

Locally finite groups and configurations

Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

<p style='text-indent:20px;'>We are concerned with the following nonlinear fractional Schrödinger equation:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ p\in\left(2+4s/N,2^*_s\right) $\end{document}</tex-math></inline-formula>, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential <inline-formula><tex-math id="M3">\begin{document}$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $\end{document}</tex-math></inline-formula>, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-normalized solution <inline-formula><tex-math id="M5">\begin{document}$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $\end{document}</tex-math></inline-formula> of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.</p>

Normalized solutions for a coupled fractional Schrödinger system in low dimensions

We consider the following coupled fractional Schrödinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in R N , with $0< s<1$ 0 < s < 1 , $2s< N\le 4s$ 2 s < N ≤ 4 s and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ 1 + 2 s N < p < N N − 2 s , under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . Assuming that the parameters $\mu _{1}$ μ 1 , $\mu _{2}$ μ 2 , $a_{1}$ a 1 , $a_{2}$ a 2 are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$ β > 0 .

Automated Normalized Solution for Finding Duplicate Records from Multiple Data Sources

Information merging may be a testing issue clinched alongside information reconciliation. The convenience of information builds when it is joined Also combined for other information from various (Web) wellsprings. The guarantee from claiming enormous information hinges upon tending to a few enormous information coordination challenges, for example, such that record linkage toward scale, ongoing information fusion, What's more coordinating profound Web. In spite of significantly fill in need been directed with respect to these problems, there may be constrained worth of effort on making An uniform, standard record from an assembly for records relating of the similar genuine world substance. Author allude with this errand as document standardization. Such a record illustration, ‘coined normalized record, may be essential for both front end and back end provisions’. In this paper, author formalize those record standardization problem, available in-depth dissection from claiming standardization granularity levels Also for standardization types. We recommend a thorough structure to registering the normalized record. Those suggested schema incorporates a suit of shield from claiming record standardization methods, from credulous ones, which utilize best the data assembled starting with records themselves, to complex strategies, which comprehensively mine an assembly about copy records when selecting a quality for a trait of a normalized record.

Complex Monge–Ampère equations on quasi-projective varieties

We introduce generalized Monge–Ampère capacities and use these to study complex Monge–Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor. Our results still hold if the divisor is replaced by any closed subset.

Complementary Method for Deriving Concentration of Diffusing Substance in a Medium for Multi-Dimensional Diffusion

ABSTRACTConcentration of a diffusing substance in a medium was derived in various cases of uni-dimensional diffusion, including in a semi-infinite medium and a plate-shaped medium. Multi-dimensional diffusion involves boundary conditions in each coordinate direction. The algorithm dealing with uni-dimensional case becomes very complicated in multi-dimensional cases. This study proposes an algorithm, which is called the complementary method, that combines complementary functions of the normalized solution in uni-dimensional diffusion case by multiplication to solve those in various multi-dimensional diffusion cases with dramatically simplified mathematics. Besides, the complementary method is used to solve various kinds of boundary conditions for multi-dimensional diffusion.