General energy bounds for systems of bosons with soft cores

We study a bound system of N identical bosons interacting by model pair potentials of the form V(r) = A sgn(p)r^p + B/r^2, A > 0, B >= 0. By using a variational trial function and the `equivalent 2-body method', we find explicit upper and lower bound formulas for the N-particle ground-state energy in arbitrary spatial dimensions d > 2 for the two cases p = 2 and p = -1. It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory

Some monotonicity results for minimizers in the calculus of variations

We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$\int F(\nabla u, u, x) dx$$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions

General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback

In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.Comment: 11 page

Some monotonicity results for general systems of nonlinear elliptic PDEs

In this paper we show that minima and stable solutions of a general energy functional of the form $$\int_{\Omega} F(\nabla u,\nabla v,u,v,x)dx$$ enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature

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