Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define $$\Delta(u)= \frac{\int u(h) \exp(-\eta u(h))\exp(-\frac{\lambda}{2}h^2)~\mathrm{d}h}{\int \exp(-\eta u(h))\exp(-\frac{\lambda}{2}h^2)~\mathrm{d}h}$$

I want to find an upper bound on $P_0$, which is defined as $$P_0=\sup_{u:\mathbb{R}\rightarrow \mathbb{R}} \Delta(u),$$ subject to the following two constraints.

- $u(h^*)=0$, for some given $h^*\in \mathbb{R}$.
- $u$ is Lipschitz in the Euclidean metric with Lipschitz constant bounded by $L$.

I have been unsuccessful at solving this problem. Here are the things that I have attempted.

Suppose we want an upper bound for the quantity. $$P_1= \max_{x_2,\ldots x_n} \frac{\sum_{i=1}^n p_i x_i \exp(-\eta x_i)}{\sum_{i=1}^n p_i \exp(-\eta x_i)}$$ where $x_1$ is fixed at 0, and the vector $[p_1,\ldots,p_n]$ is a probability vector, i.e. $p_i\geq 0,\sum_{i=1}^n p_i=1$. The maxima for the optimization problem involved in the calculation of $P_1$, is attained when $x_2=\ldots=x_n=x$. One can then show, via straightforward calculations, that $P_1\leq \frac{1}{\eta}\log(1/p_1)$. Calculation of an upper bound on $P_0$, can be seen as something similar to calculating an upper bound on $P_1$, just that we now have a continuous problem.I am guessing that the following function $u_0(h)$ might be the optimizer of $\Delta(u)$.

$$ u_0(h)= \begin{cases} a_1(h-h^{*})~\text{if}~ h^{*}\leq h\leq h_{+}\\ a_1(h_{+}-h^{*})~\text{if}~ h\geq h_{+}\\ a_2(h-h^{*})~\text{if}~ h_{-}\leq h\leq h^{*}\\ a_2(h_{-}-h^{*})~\text{if}~ h\leq h_{-}, \end{cases} $$ where $0\leq a_1\leq L,-L\leq a_2\leq 0$, are appropriate constants. This way I can mimic the behavior of the solution of problem $P_1$, for our continuous problem. However, I have no idea as to how to prove that $u_0(h)$ is the optimal solution for $P_0$.

A second attempt, would be to write the individual integrals in the numerator and denominator of $\Delta(u)$ as discrete summation, and then pass to the limit. The advantage of doing this is that we immediately convert our problem into a discrete problem, and then utilize what we know about upper bounding $P_1$, in order to get an upper bound on $P_0$. However I have not been able to implement this technique, and I would deeply appreciate, if one could comment, if such a technique is valid and sketch the main details.

P.S. I have a feeling that this problem is not hard, and perhaps has been studied before. However, I was neither able to get a reference, nor was able to solve it by myself.