We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.
In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Liebs Translation Lemma and a Riesz energy version of the Brezis--Lieb lemma.
We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.
We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.
For the BCS equation with local two-body interaction $lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as $lambda to 0$. We derive necessary and sufficient conditions on $V(x)$ for the existence of a non-trivial solution for all values of $lambda>0$.
We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Polyas conjecture is not true in the presence of a magnetic field.
