Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed $L2$ norm for the Schrodinger equation $$-\Delta u+\lambda u+V(x)u=|u|{p-2}u,\quad u\in H1_0(\Omega),\quad\int_\Omega u2{\rm d}\,x=\rho2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}N$ or $\mathbb {R}N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (D?), or if $\|V\|_{Lq}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of $\mathbb {R}N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}N$

Quelques aspects de la petite irrigation dans le Nordeste brésilien

Le Sertao, zone semi-aride du Nordeste brésilien, à pluviométrie très irrégulière, est confronté de manière cruciale au problème du stockage et de l'utilisation des ressources en eau. Longtemps zone d'élevage et d'agriculture pluviale de subsistance, le Sertao connaît, depuis une dizaine d'années, une importante vague d'expansion de l'irrigation : cette évolution remet en cause la valeur symbolique des ressources en eau (réserve pour l'alimentation en cas de sécheresse) et favorise l'émergence de nouveaux systèmes de production. Les auteurs font le point sur les diverses composantes de ce développement et sur les principaux problèmes qu'ils rencontrent. Ils résument les travaux de la Coopération Française et de l'ORSTOM à ce sujet. (Résumé d'auteur

Normalized solutions to mass supercritical Schroedinger equations with negative potential

We study the existence of positive solutions with prescribed L-2-norm for the mass supercritical Schrodinger equation -delta u+lambda u - V(x)u = |u|(p-2)u u is an element of H-1(R-N), lambda is an element of R, where V >= 0, N >= 1 and p is an element of(2+4/N, 2*), 2*: = 2N/N-2 if N >= 3 and 2* : = +infinity if N = 1,2. We treat two cases. Firstly, under an explicit smallness assumption on V and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on V, and that V is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved

Multiple positive bound state solutions for a critical Choquard equation

In this paper we consider the problem(P lambda()) {-Delta u + V-lambda(x)u = (I-mu * vertical bar u vertical bar(2 mu*))vertical bar u vertical bar(2 mu*-2)u in R-N,u > 0 in R-N,where V-lambda = lambda + V-0 with lambda >= 0, V-0 is an element of L-N/2(R-N), I-mu = 1/vertical bar x vertical bar(mu) is the Riesz potential with 0 < mu < min{N, 4} and 2(mu)* = 2N-mu/N-2 with N >= 3. Under some smallness assumption on V-0 and lambda we prove the existence of two positive solutions of (P-lambda). In order to prove the main results, we used variational methods combined with degree theory

Unconventional Magnetization below 25 K in Nitrogen-doped Diamond provides hints for the existence of Superconductivity and Superparamagnetism

The magnetization of nitrogen-doped single crystalline diamond bulk samples shows unconventional field and temperature hysteresis loops at T ≲ 25 K. The results suggest the existence of superparamagnetic and superconducting regions in samples with nitrogen concentration <200 ppm. Both phases vanish at temperatures above 25 K where the samples show diamagnetic behavior similar to undoped diamond. The observation of superparamagnetism and superconductivity is attributed to the nitrogen doping and to the existence of defective regions. From particle-induced X-ray emission with ppm resolution we rule out that the main observations below 25 K are due to magnetic impurities. We investigated also the magnetic properties of ferromagnetic/high-temperature superconducting oxide bilayers. The magnetization results obtained from those bilayers show remarkable similarities to the ones in nitrogen-doped diamond