Let $\Omega$ be a Polish space with its Borel $\sigma$ algebra and a non atomic probability measure.
Let $$U : L^2(\Omega, \mathbb{R}^d) \to \mathbb{R}$$ be a $C^1$ function (in Frechèt sense). Let $X \in L^2(\Omega, \mathbb{R}^d)$ be a fixed r.v. Consider the following functional $$ F_{\epsilon, X} (Y)= U(Y) + \frac{1}{2\epsilon} \mathbb{E}[|Y-X|^2] + 2\mathbb{E}[|Y|^4]$$ and the minimization problem $$\min_{Y \in L^4(\Omega, \mathbb{R}^d)} F_{\epsilon,X}(Y)$$ Since $U$ is differentiabile at $X$ there exists a ball of radius $r>0$ of $X$, call it $B$, where $U$ is equibounded.
Am I able to say that for a sufficiently small $\epsilon$ every minimizing sequence of $F_{\epsilon,X}$ is eventually inside $B$?
Without any further hypothesis on $U$ the general answer to my question is "no". Indeed take $X=0 \in \mathbb{R}^d$ and $$U(Y)= -\mathbb{E}[|Y|^2]^3$$ for every $Y \in L^2(\Omega, \mathbb{R}^d)$. $U$ is clearly continuous. Now take $Y_n = n$ for each $n \in \mathbb{N}$, then this is a minimizing sequence for each $F_{\epsilon, 0}$: $$F_{\epsilon,0}(Y_n) = -n^6+\frac{n^2}{2\epsilon}+2n^4 \overset{n \to +\infty}{\longrightarrow} - \infty$$ but clearly $\{Y_n\}_{n \in \mathbb{N}}$ is not eventually contained in any ball of positive radius centered at $X=0$.