Strong Maximum Principle
Let $\Omega$ be an open connected bounded set. Suppose $u\in C^2(\Omega)\cap C(\bar \Omega)$ satisfies
$$\Delta u + c(x) u\geq0$$ where $c\in C(\Omega)$ and $c(x) \leq 0$, for all $x\in \Omega$. Suppose $M:=\sup\{u(x),x\in \Omega\} > 0.$ Then $u$ has maximum on boundary of $\Omega$.
My problem is
Let $\Omega=(0,1)^2$. Suppose $u \in C^2(\Omega) \cap C(\bar \Omega)$ satisfies $$\Delta u - u^3=0 \mbox{ in $\Omega$ }$$ $$u(x,1)=0,u(x,0)=1$$ Show that $u$ is unique and $0\leq u \leq 1$.
My try :
Let $u_1,u_2$ be solutions of this equation.
Let $w=u_1-u_2$ By easy calculation,
$$\Delta w -(w^2+3u_1u_2)w=0$$ and $$c=-(w^2+3u_1u_2) \leq 0$$
so
$$\Delta w +cw \geq 0$$ $$w(x,1)=0,w(x,0)=0$$
I think I can use Strong Maximum Principle.
But I think I need some conditions $$ w(0,y)=w(1,y) =0$$
If this conditions is satisfied, I'm done.
Do you think given conditions in this problem are sufficient to solve it?