Let $f_n \in C^2(\bar{\Omega})$ be a sequence satisfying

$\Delta f_n - f_n^3 \to 0 \ \ {\rm in} \ \ L^2(\Omega)$

where $\Omega \subset {\mathbb R}^2$ is bounded and open with a smooth boundary. Is it necessarily true that $\|f_n^3\|_{L^2(\Omega)}$ and $\|\Delta f_n\|_{L^2(\Omega)}$ are uniformly bounded? If not, can you give a counterexample?

If there is a counterexample, I imagine it would involve $f_n$ becoming unbounded on $\partial \Omega$. If so, is it possible that this statement is true for $f_n \in C^2_c(\bar{\Omega})$ (i.e. if $f_n$ all have compact support in $\Omega$.)

The reason I believe it may be true is that the condition $\Delta f_n - f_n^3 \to 0$ seems incompatible with the sequence $f_n$ having an increasing positive interior maximum or a decreasing negative interior minimium as $\Delta f_n$ and $-f_n^3$ would have the same signs and not cancel out. I have tried to come up with counterexamples in one dimension but have had no luck yet.