Let $\lambda\in\mathbb{R}$, $2<p\leq 2^*=(2N)/(N-2)$. For $u \in H^{1}_0({\Omega})$ where $\Omega$ is a domain of $R^N$. Define:
$$ \varphi(u) = \displaystyle\int_{\Omega} \frac{1}{2}|\nabla u|^2 +\frac{\lambda}{2}u^2 - \frac{1}{p}|u|^p\ dx$$
I am trying to show that the supremum of $\varphi$ in $H_0^1(\Omega)$ is $\infty$. Thus I tried to prove it is coercive, that is $\varphi(u) \rightarrow \infty$ as $|| u||_{H^{1}_{0}} \rightarrow \infty$. I have tried to use the G-N inequality and Sobolev equality, but the third term is of higher power, and I don't know how to estimate this term.
Besides, maybe this functional is not coercive? But how to prove that it is unbounded from above?
I'll appreciated it for any hints and answers!
I believe the following argument works:
Suppose $0\in \Omega$ for simplicity. Given a positive, smooth bump $u$ with support in $\Omega$, and $0<r<1$ consider $u_r(x_1, \ldots, x_n)=u(x_1/r,x_2\ldots, x_n)$; in other words, we scale $u$ in the first coordinate and leave the others unchanged. Then by a change of variables (in the first term only!), and writing all integrals over $\mathbb{R}^n$ owing to the support properties of $u$, $$ \varphi(u_r)\geq \int_{\mathbb{R}^n}\dfrac{1}{2r} |\partial_{x_1} u(x)|^2\, dx + \int_{\mathbb{R}^n} \dfrac{\lambda}{2}u_r^2 - \dfrac{1}{p} |u_r|^p\, dx $$
The last two terms on the right converge to 0, as $r\to 0$, by dominated convergence since the support of $u_r$ is shrinking towards the plane $x_1=0$. Therefore $\varphi(u_r)\geq (2r)^{-1}\| \partial_{x_1} u\|_2^2+ o(1)$ as $r\to 0$.