Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > 0$, $\left\| F \right\|_{C^1(\left(-\infty,0\right])} < \infty$, $F$ is asymptotically 0, $F > 0$ for $y\in \left(-\infty,0\right)$, and $F(0) = 0$.
Let $\psi$ be a minimizer. My question is, can we say anything about $\psi(0)$?
Edit: I should also state that $c < m(y) < C$ for positive constants $c,C$.