The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model

The thermal equilibrium state of a bipolar, isothermic quantum fluid confined to a bounded domain Omega subset of R-d, d = 1,2 or d = 3 is entirely described by the particle densities n, p, minimizing the energy epsilon(2) integral \del root n\(2) + epsilon(2) integral \del root p\(2) + integral G(1)(n) + integral G(2)(p) + lambda(2)/2 integral \del V[n-p-C]\(2), where G(1,2) are strictly convex real valued functions, -lambda(2) Delta V = n-p-C, with integral(n-p-C) = integral V = 0. It is shown that this variational problem has a unique minimizer in {(n, p) is an element of L-1(Omega) x L-1(Omega); n, p greater than or equal to 0, root n, root p is an element of H-1(Omega), integral n = N, integral p = P} and some regularity results are proven. The semi-classical limit epsilon --> 0 is carried out recovering the minimizer of the limiting functional. The subsequent zero space charge limit lambda --> 0 leads to extensions of the classical boundary conditions. Due to the lack of regularity the asymptotics lambda --> 0 can not be settled on Sobolev embedding arguments. The limit is carried out by means of a compactness-by-convexity principle.