Estimate a function given an estimate of its Laplacian

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 $f_\lambda$?

• I suggest trying $f(x)=c(e^{ax}-e^{-ax})$ in $[-1,1]$ (if you didnt already...) – Jean Duchon Apr 11 '18 at 16:58
• Just use the radial symmetrization to reduce it to a $1$-dimensional problem for the value estimates. The uniform estimates for the derivatives are a no-go because $|x|^\alpha$ is subharmonic for $n\ge 2$ and any $\alpha>0$, so you can always add a small multiple of it to spoil the smoothness (or the Holder condition) at the origin. – fedja Apr 11 '18 at 18:09