Let $\Omega \subset \mathbb{R}^n$ be a bounded $C^1$-domain and $\varphi$ be a continuos function on $\partial\Omega$. Let $u\in C^2(\Omega ) \cap C^1(\bar{\Omega})$ a solution of the problem:
$$\Delta u -u^3=f\qquad \text{in}\quad\Omega$$ $$u =\varphi\qquad \text{in}\quad\partial \Omega$$
I want to show the uniqueness of the solution.
I try to use the energy method. I considered $u_1$ and $u_2$ two solutions of the problem and then I made $w=u_1-u_2$. So I hade the following problem:
$$\Delta w -u_1^3+u_2^3=0\qquad \text{in}\quad\Omega$$ $$w =0\qquad \text{in}\quad\partial \Omega$$
I saw that $u_1^3-u_2^3=w(u_1^2+u_1u_2+u_2^2)$. And then, after multiplying the first equation by $w$ and after integrate in $\Omega$, I got:
$$\int_{\Omega}|\nabla w|^2 dx + \int_{\Omega}w^2(u_1^2+u_1u_2+u_2^2) dx=0$$
My question now is: Since $u_1^2+u_2^2\geq\frac{u_1^2+u_2^2}{2}\geq u_1u_2$, can I conclude that both integrals are equal to zero, because they are greater or equal to zero or do I need to prove anything else or is there another way to prove the uniqueness of the solution?
Thank you.
