Integral inequality in $L^p(\Omega)$

Question

Let $\Omega$ be an open bounded set and $u\in L^p(\Omega)$, with $p>1$. Let $\Omega_1 = \left\lbrace x\in\Omega\mid u(x) >k\right\rbrace$ where $k$ denotes a positive constant greater than 1. It is always true that $$\int_{\Omega_{1}} \vert u\vert^p dx >1?$$

Any hints to prove (or not!) that it is true?

Thank You!

Answer 1

One easily finds counterexamples trivial counterexamples: $u=0\in L^p$

But there are also quite easy non-trivial counterexamples, e.g. using characteristic functions of "small enough" sets or functions bounded by $1$.