Why $J(u)$ satisfies mountain pass geometry?

Question

Let $\Omega\subset \mathbb{R}^N$ be a smooth domain and $W_0^{1,p}(\Omega)$ be standard Sobolev space. Define the variational functional by $$J(u)=\frac{1}{p}\int_\Omega|\nabla u|^pdx-\frac{1}{q}\int_\Omega|u|^qdx,\quad u\in W_0^{1,p}(\Omega)$$ I want to varify that $J(u)$ satisfies Mountain Pass Geometry if $p<q<\frac{Np}{N-p}$

1. $J(0)=0$ is obvious.
2. there exists constant $r,a>0$ such that $J(u)\geq a$ if $||u||=r$

I try to use poincare inequality. $J(u)\geq\frac{1}{p}||\nabla u||^p_{L_p}-C||\nabla u||^q_{L_p}$. If $r$ is sufficiently sall $J(u)$ can larger than a constant $a$.

3. There exists an element $v$ with $||v||>r,J(u)\leq0$.

My question is how to verify the third condition?