Duality and best constant for a Trudinger-Moser inequality involving probability measures

We consider the Trudinger-Moser type functional \[ J_\lambda(v)=\frac{1}{2}\int_\Omega|\nabla v|^2-\lambda\int_I\left(\log\int_\Omega e^{\alpha v}\right)\,\calP(d\alpha), \] where $\Omega$ is a two-dimensional Riemannian surface without boundary, $v\in H^1(\Omega)$, $\int_\Omega v=0$, $I=[-1,1]$, $\calP$ is a Borel probability measure on $I$ and $\lambda>0$. The functional $J_\lambda$ arises in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by $\calP$. We formulate a Toland non-convex duality principle for $J_\lambda$ and we compute the optimal value of $\lambda$ for which $J_\lambda$ is bounded from below.