SUPERLINEAR ELLIPTIC PROBLEMS WITH SIGN CHANGING COEFFICIENTS

Via variational methods, we study multiplicity of solutions for the problem {-Delta u = lambda b(x)vertical bar u vertical bar(q-2)u + au + g(x, u) in Omega, u - 0 on partial derivative Omega, where a simple example for g( x, u) is |u|(p-2)u; here a, lambda are real parameters, 1 < q < 2 < p <= 2* and b(x) is a function in a suitable space L-sigma. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any lambda > 0, and a total of five nontrivial solutions are obtained when lambda is small and a >= lambda(1). Note that this type of results are valid even in the critical case.

Long-term leucine supplementation reduces fat mass gain without changing body protein status of aging rats

Objective: Aging is characterized by alterations in body composition such as an increase in body fat and decreases in muscle mass (sarcopenia) and bone density (osteopenia). Leucine supplementation has been shown to acutely stimulate protein synthesis and to decrease body fat. However, the long-term effect of consistent leucine supplementation is not well defined. This study investigated the effect of leucine supplementation during aging. Methods: Six-month-old rats were divided into three groups: an adult group (n = 10) euthanized at 6 mo of age, a leucine group (n = 16) that received a diet supplemented with 4% leucine for 40 wk, and a control group (n = 19) that received the control diet for 40 wk. The following parameters were evaluated: body weight, food intake, chemical carcass composition, indicators of acquired chronic diseases, and indicators of protein nutritional status. Results: Body weight and fat were lower in the leucine group after 40 wk of supplementation compared with the control group but still higher than in the adult group. The lipid and glycemic profiles were equally altered in the control and leucine groups because of aging. In addition, leucine supplementation did not affect the changes in protein status parameters associated with aging, such as decreases in body and muscle protein and total serum protein. Conclusion: The results indicate that leucine supplementation attenuates body fat gain during aging but does not affect risk indicators of acquired chronic diseases. Furthermore, supplemented animals did not show signs of a prevention of the decrease in lean mass associated with aging. (C) 2012 Elsevier Inc. All rights reserved.