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There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C 2 C^2 -domain in R n \mathbb {R}^n of the following form ∫ Ω d β ( x ) | ∇ u ( x ) | 2 d x ≥ C ( α , β ) ∫ Ω | u ( x ) | 2 d α ( x ) d x with ∫ Ω u ( x ) d α ( x ) d x = 0 , \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d ( x ) d(x) is the distance from x ∈ Ω x \in \Omega to the boundary ∂ Ω \partial \Omega and α , β ∈ R \alpha ,\beta \in \mathbb {R} . We classify all ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 for which C ( α , β ) > 0 C(\alpha ,\beta ) > 0 . Then, we study whether an optimal constant C ( α , β ) C(\alpha ,\beta ) is attained or not. Our study on C ( α , β ) C(\alpha ,\beta ) for general ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version.

This paper is concerned with asymptotic behavior of positive solutions for coupled Schrödinger equations with mixed interactions between components. We construct locally minimal energy solutions that show distinctively different limiting profile for simultaneously large attractive and repulsive couplings. The components of the solutions constructed exhibit partial synchronization and segregation.