Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions: \begin{eqnarray*} \int_B |f_\lambda(x)|^2dx\leq 1,\\ \Delta f_\lambda\geq \lambda \cdot f_\lambda, \end{eqnarray*} where $\Delta =\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian.

Is it true that $||f_\lambda||_{C^0(\frac{1}{2} B)}\to 0$ as $\lambda \to +\infty$? If yes, is it possible to give more precise estimates on the rate of convergence and possibly of the derivatives of $f_\lambda$?