If $\|u\|_{2}<\infty $ then $\|u\|_{p}<\infty (p>2)$?

Question

My question is: Suppose that $\|u\|_{2}<\sqrt{c} \quad (u:\Omega\rightarrow R) $ and $\Omega$ is an open bounded domain in $R^{n}$ . Is it possible to say that $\|u\|_{p}<\infty (p>2)$ ? My attempt: By $\|u\|_{2}<\sqrt{c}\ $ we then have $|u|<\sqrt{\frac{c}{|\Omega|}}$ (since if $|u|\geq\sqrt{\frac{c}{|\Omega|}}$ then $\|u\|_{2}\geq\sqrt{c}$). Therefore, $|u|^{p}<\left(\sqrt{\frac{c}{|\Omega|}}\right)^p$ and then integrating over $\Omega$ we get $\|u\|_{p}<\infty$. Am I right or I am missing something?

Answer 1

No. Your conclusion that $|u|\le \sqrt{c/|\Omega|}$ is incorrect. Consider $u(x)=x^a$ in $\Omega=(0,1)$ with $0<a<1/2$.