Prove properties about equation $-u’’+u=u^p$

Question

Assume that $p>1,u\ge0,u\in H^1(\mathbb R),u(0)=\Vert u\Vert_{L^\infty}$, and $u$ satisfies that $$-u’’+u=u^p$$ Prove that $u(0)>1,u$ is even and is strictly decreasing on $[0,+\infty)$ $$$$ It is easy to prove $u$ is smooth, but I didn’t figure out about the decreasing property. It seems that we can use maximum principle or something but I don’t know how to do. Are there any suggestions about this problem?