Splitting the Fučík Spectrum and the Number of Solutions to a Quasilinear ODE

For $\varnothing$ an increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ and $f\epsilon C\left(\mathbb{R}\right)$, we consider the problem $$\left(\varnothing\left(u'\right)\right)'+f\left(u\right)=0,\qquad t\epsilon\left(0,L\right),\qquad u\left(0\right)=0=u\left(L\right).$$ The aim is to study multiplicity of solutions by means of some generalized Pseudo Fu$\check{\textrm{c}}$ik spectrum (at infinity, or at zero). New insights that lead to a very precise counting of solutions are obtained by splitting these spectra into two parts, called Positive Pseudo Fu$\check{\textrm{c}}$ik Spectrum (PPFS) and Negative Pseudo Fu$\check{\textrm{c}}$ik spectrum (NPFS) (at infinity, or at zero, respectively), in this form tue can discuss separately the two cases u' (0) > 0 and u' (0) < 0

On the uniqueness of sign changing bound state solutions of a semilinear equation

We establish the uniqueness of the higher radial bound state solutions of \Delta u +f(u)=0,\quad x\in \RR^n. \leqno(P) We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth conditions, and either has one zero at $b>0$, is non positive and not identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or is positive and differentiable in $[0,\infty)$

Folate status in european adolescents and its relation to obesity

En la actualidad la prevalencia de obesidad en adolescentes va en aumento. Recientemente algunos estudios han mostrado una relación entre las concentraciones de folato y la obesidad. Este hecho relevante ya que el folato participa en diversas rutas metabólicas en el organismo. Por ello este trabajo trata de identificar los factores que afectan a los niveles de folato y a estudiar su relación con la obesidad. Se examinaron datos de participantes del estudio multicentrico HELENA (n=1089; 12.5-17.5 años; 580 niñas). Se estudiaron las concentraciones de folato en plasma (PF) y eritrocitos (RBCf), según variables biológicas, conductuales y de obesidad; índice de masa corporal (BMI), índice cintura talla (WHtr) circunferencia de la cintura (WC), pliegue tricipital (TSK) y porcentaje de grasa corporal (BF%). Sólo el 17.4% fue deficiente para el PF, y el 3.3% para el RBCf. Más del 50% de la población está por debajo de las necesidades medias estimadas (EAR). La asociación entre el estado de folato y la ingesta de folato apenas tiene impacto clínico, y no se da en individuos con sobrepeso y obesidad. La edad se asocia con un descenso de los niveles de PF y el estado puberal con una disminución de los niveles de PF y RBCf. Los portadores del alelo TT para la mutación de la C677T MTHFr tuvieron los menores niveles de PF, pero los mayores de RBCf. No se encontró ninguna relación entre los niveles de folato y el estado de obesidad.ABSTRACT: In the current moment the prevalence of adolescent obesity is increasing. Recently several studies have shown a relation between folate status and obesity. This fact is relevant taking into account that folate is involved in a wide number of metabolic pathways in the organism. Thus, this study tries to identify factor affecting folate levels and its relation to obesity. Data coming from multicentric study HELENA were examined (n=1089; 12.5-17.5y; 580 female). Plasma folate (PF) and red blood cell folate (RBCf) levels were studied acording to biological and conductual variables and also obestiy considering body mass index (BMI), waist to height ratio (WHtr), waist circumference (WC), tricipital skinfold thickness (TSK) and body fat porcentaje (BF%). Only the 17.4% was deficient for PF, and the 3.3% for RBCf. More tan the 50% was under the estimated average requierement (EAR). The association between foalte status and intake did not show an important clinical impact, and we did not even observe it in overweight and obese subjects. Age is linked to impaired PF levels, and puberal status to impaired PF and RBCf levels. TT carriers for C677T MTHFr had the lowest PF levels but the highest RBCf levels. We did not find any relation between foalte status and any of the obesity variables.Máster en Condicionantes Genéticos, Nutricionales y Ambientales del Crecimiento y el Desarroll

Existence of positive radial solutions for a weakly coupled system via blow up

The existence of positive solutions to certain systems of ordinary differential equations is studied. Particular forms of these systems are satisfied by radial solutions of associated partial differential equations

Positive solutions for systems of quasilinear equations with non-homogeneous operators and weights

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [12], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [8], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray-Schauder topological degree theory.Comisión Nacional de Investigación Cientifica y Tecnológica (CONICYT) CONICYT FONDECYT 1160540 Conicyt Basal Grant PIA PFB170001 Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI) 26400182 19K03595 17H0109