Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the following lower semi continuous (w.r.t weak topology) strictly convex functional $$E(\mu)=\int_\mathbb{R}V_t(x)d\mu(x)-P\iint_{\mathbb{R}^2}\log|x-y|d\mu(x)d\mu(y)+\int_\mathbb{R}\log\Big(\dfrac{d\mu}{dx}\Big)d\mu(x)$$
with respect to $t$ and $P>0$. The minimizer $\mu_t$ of this equation with p.d.f $\rho_t$ is characterized by the following equation:
There exists a constant $C_t^P\in\mathbb{R}$ such that for all $x\in\mathbb{R}$ $$V_t(x)-2P\int_\mathbb{R}\log|x-y|\rho_t(y)dy+\log\rho_t(x)=C_t^P$$.
We can differentiate w.r.t $x$ and find the relation $O[\rho_t]=-V'_t$ where $O$ is defined by $$O[f]=\dfrac{f'}{f}+2P\mathcal{H}[f]$$
where $\mathcal{H}[f](x):=p.v\displaystyle\int_\mathbb{R}\dfrac{f(t)}{t-x}dt$.
Is there a small chance that $\rho_t$ will be continuous with respect to $t$ ? I think it would be easier to invert the non-linear operator $O$ and show continuity of the inverse with small $P$.