Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by:
$$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where $p <\frac{2n}{n-2}$ I want to show that $J$ is coercive, i.e $\lim_{\|v\|_{\mathscr{H}^{1}_0(\Omega)} \to \infty} |J(v)| = \infty$ to set up a later maximization problem. I have tried using a host of inequalities (Sobolev, Holder interpolation) to show the above limit, but they have all failed. The main issue occurs when the $L^2$ norm of $\nabla u$ grows unboundedly, but the $L^2$ norm of $u$ does not. Of course, the $L^2$ norm of $u$ can be controlled above by the norm of its gradient (Poincare inequality), but this does not give us control of the norms from below, which is what I seek.
This is not true. Let $\Omega$ be the unit disk in $\mathbb{R}^2$ (you can construct similar examples in higher dimensions), $\Omega'=(-R,R)^2$ for some $R<1$, $\eta\in C_c^\infty(\Omega)$ with $0\leq \eta\leq 1$, $\eta(x)=1$ for $x\in \Omega'$, and $$ v_\epsilon(x_1,x_2)=\begin{cases}-1&\text{if }x_1\leq -\epsilon,\\ x/\epsilon&\text{if }-\epsilon<x_1<\epsilon,\\ 1&\text{if }x_1\geq \epsilon.\end{cases} $$ Then $\eta v_\epsilon\in H^1_0(\Omega)$, $\|\eta v_\epsilon\|_p\leq |\Omega|^{1/p}\|\eta v_\epsilon\|_\infty\leq |\Omega|^{1/p}$, yet $$ \int_{\Omega}|\partial_1(\eta v_\epsilon)|^2\,dx\geq \int_{\Omega'}|\partial_1 v_\epsilon|^2\,dx=\int_{-R}^R \int_{-\epsilon}^{\epsilon}\epsilon^{-2}\,dx_1\,dx_2=4R\epsilon^{-1}\to\infty. $$