I wish to prove that the following PDE has at most one solution
\begin{align} \Delta u - u^3 &= f, \quad x \in \Omega \\ u &= \phi, \quad x \in \partial \Omega \end{align}
with $f$ continuous in $\Omega$. I tried to apply the energy method, but it didn't work.
I started by supposing that there are two solutions, $u,v$. Then I set $w = u - v$. Therefore, we have that
$$\Delta w = u^3 - v^3$$
where $w$ equals to zero on the boundary. I tried multiplying by w and integrating over $\Omega$. Then I got
$$\|\nabla w\|^{2}_{L^{2}} = \|u^{2} + uv + v^{2}\| \|w\|^{2}_{L^{2}}$$
Then, I got stuck.
I appreciate any help. Thank you.