Uniqueness of a nonlinear PDE

Question

I have a nonlinear PDE inside with a domain of the unit ball: $$\begin{cases} \Delta u = u^3 &\text{in} \ \ B(x,1) \\ u =0 &\text{on} \ \ \partial B(x,1) \end{cases} $$ I am asked to show uniqueness of this problem. Up to this point, I have only shown uniqueness and existence for linear PDE, not nonlinear PDE. Usually, the proof is as easy as multiplying by $u$, applying the divergence theorem, and the identity $D(uDu) = u\Delta u + |Du|^2$. However, since the PDE is nonlinear, this does not work. It would be great if someone could help me through this, or provide a reference to aid me.

Answer 1

You are right that the standard method does not work as it relies on linearity, but the same technique can be applied. Mutiplying the PDE by $u$ and integrating over $B(x,1)$ using $-u\Delta u = |\nabla u|^2 - \nabla\cdot(u\nabla u)$ and applying the divergence theorem to the last term gives us $$u^4 - u\Delta u = 0 \implies \underbrace{\int [u^4 + |\nabla u|^2]{\rm d}V}_{>0~\text{if $u\not\equiv 0$ since integrand is positive}} - \underbrace{\int u\nabla u\cdot {\rm d}S}_{\text{zero since}~u =0~\text{on boundary}} = 0$$

The only way this can hold true is if $u^4 + |\nabla u|^2 \equiv 0$ so $u \equiv 0$. It's also easy to check that this is indeed a solution to the problem.