Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

in this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary a deltaOhm and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1). (C) 2003 Published by Elsevier Inc.

Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

Hypoxic/ischemic models in newborn piglet: Comparison of constant FiO(2) versus variable FiO(2) delivery

A comparison of a constant (continuous delivery of 4% FiO(2)) and a variable (initial 5% FiO(2) with adjustments to induce low amplitude EEG (LAEEG) and hypotension) hypoxic/ischemic insult was performed to determine which insult was more effective in producing a consistent degree of survivable neuropathological damage in a newborn piglet model of perinatal asphyxia. We also examined which physiological responses contributed to this outcome. Thirty-nine 1-day-old piglets were subjected to either a constant hypoxic/ischemic insult of 30- to 37-min duration or a variable hypoxic/ischemic insult of 30-min low peak amplitude EEG (LAEEG < 5 mu V) including 10 min of low mean arterial blood pressure (MABP < 70% of baseline). Control animals (n = 6) received 21% FiO(2) for the duration of the experiment. At 72 h, the piglets were euthanased, their brains removed and fixed in 4% paraformaldehyde and assessed for hypoxic/ischemic injury by histological analysis. Based on neuropathology scores, piglets were grouped as undamaged or damaged; piglets that did not survive to 72 h were grouped separately as dead. The variable insult resulted in a greater number of piglets with neuropathological damage (undamaged = 12.5%, damaged = 68.75%, dead = 18.75%) while the constant insult resulted in a large proportion of undamaged piglets (undamaged = 50%, damaged = 22.2%, dead = 27.8%). A hypoxic insult varied to maintain peak amplitude EEG < 5 mu V results in a greater number of survivors with a consistent degree of neuropathological damage than a constant hypoxic insult. Physiological variables MABP, LAEEG, pH and arterial base excess were found to be significantly associated with neuropathological outcome. (c) 2006 Elsevier B.V. All rights reserved.

Muscle metabolism during constant- and alternating-intensity exercise around critical power

Few studies have focused on the metabolic responses to alternating high- and low-intensity exercise and, specifically, compared these responses to those seen during constant-load exercise performed at the same average power output. This study compared muscle metabolic responses between two patterns of exercise during which the intensity was either constant and just below critical power (CP) or that oscillated above and below CP. Six trained males (mean +/- SD age 23.6 +/- 2.6 y) completed two 30-minute bouts of cycling (alternating and constant) at an average intensity equal to 90% of CR The intensity during alternating exercise varied between 158% CP and 73% CP. Biopsy samples from the vastus lateralis muscle were taken before (PRE), at the midpoint and end (POST) of exercise and analysed for glycogen, lactate, PCr and pH. Although these metabolic variables in muscle changed significantly during both patterns of exercise, there were no significant differences (p > 0.05) between constant and alternating exercise for glycogen (PRE: 418.8 +/- 85 vs. 444.3 +/- 70; POST: 220.5 +/- 59 vs. 259.5 +/- 126mmol.kg(-1) dw), lactate (PRE: 8.5 +/- 7.7 vs. 8.5 +/- 8.3; POST: 49.9 +/- 19.0 vs. 42.6 +/- 26.6 mmol.kg(-1)dw), phosphocreatine (PRE: 77.9 +/- 11.6 vs. 75.7 +/- 16.9; POST: 65.8 +/- 12.1 vs. 61.2 +/- 12.7mmol.kg(-1)dw) or pH (PRE: 6.99 +/- 0.12 vs. 6.99 +/- 0.08; POST: 6.86 +/- 0.13 vs. 6.85 +/- 0.06), respectively. There were also no significant differences in blood lactate responses to the two patterns of exercise. These data suggest that, when the average power output is similar, large variations in exercise intensity exert no significant effect on muscle metabolism.