Does Lebesgue Dominated Convergence Theorem apply? Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $(v_n)_n\subset W_0^{1, p}(\Omega)$ with $p > 1$. Moreover, let $k>1$ a positive constant and consider
$$\Omega_{n, k}:=\left\lbrace x\in\Omega\mid v_n(x) > k\right\rbrace.$$
If $\psi$ is a real-valued function such that $\psi(v_n)\longrightarrow 0$, it is true that
$$\int_{\Omega\setminus\Omega_{n, k}} \psi(v_n) \vert\nabla v_n\vert^{p} dx \longrightarrow 0?$$
I think that (maybe) Lebesgue Dominated Convergence Theorem applies, but could anyone help me in understing why?
Thank you in advance!
 A: The following counterexample is a verification of my guess in the comments. As expected, $v_n$ being small has very little to do with $\nabla v_n$ being small, and the pointwise decay of $\psi(v_n)$ does not mean that the product $\psi(v_n)|\nabla v_n|^p$ decays to $0$ pointwise.
Let $0<\epsilon\ll 1$, and let $w\ge 0$ be a bump function compactly supported in $\Omega\subset \mathbb R^d$ for $d\ge 1$, where (W.L.O.G.) there is a set $U=[-a,a]\times U'\Subset \Omega$ where $w(x)\equiv 1$,  $|U|>0$, $\|w\|_{L^\infty}=1$ and define for $n\ge1$,
$$ s_n(x) := \sin(nx_1), \quad c_n(x) :=  \cos(nx_1),$$
$$ v_n(x) := n^{-\epsilon}w(x)\left(\frac12+\frac14s_n(x)\right) \in C^\infty_0(\Omega),$$
$$ \psi(v):= v. $$
Here, $x_1 := \vec e_1 \cdot x$ is the first component of $x$. Then $0\le v_n(x) \le \|v_n\|_{L^\infty}\le n^{-\epsilon} \le 1 < k$, so $\Omega_{n,k}=\emptyset$. This also proves $\psi(v_n)\to 0$ a.e. (actually, convergence is uniform). Also $\partial_1 v_n = 4^{-1}n^{-\epsilon} \partial_1w(2+s_n) + 4^{-1}n^{1-\epsilon} w c_n. $ This gives
\begin{align}
 \int_{\Omega \setminus \Omega_{n,k} } \psi(v_n)|\nabla v_n|^p 
&\ge \int_{\mathbb R^d} n^{-\epsilon} w\left(\frac12 + \frac14 s_n\right)|\partial_1v_n|^p 
\\&= n^{p-\epsilon(p+1)}\int_{\mathbb R^d}w\left(\frac12 + \frac14 s_n\right)\left|\frac{\partial_1 w\left(\frac12 + \frac14 s_n\right)}{n} + \frac{wc_n}4\right|^p \end{align}
Restricting now to the set $U$ where $w\equiv 1$, and therefore $\partial_1 w \equiv 0$,
$$ \int_{\Omega \setminus \Omega_{n,k} } \psi(v_n)|\nabla v_n|^p  \gtrsim_{w,p} n^{p-\epsilon(p+1)}\int_{-a}^{+a} |\cos(n s)|^p ds $$
It shouldn't be hard to show using the periodicity of $\cos$ that $$ \int_{-a}^a |\cos(n s)|^p ds = \frac1n\int_{-na}^{+na}|\cos t|^p dt$$  converges to some positive constant $C_{a,p}$. The upshot is
$$  \int_{\Omega \setminus \Omega_{n,k} } \psi(v_n)|\nabla v_n|^p  \to \infty .$$
The integrand doesn't converge pointwise to 0 (and there is of course no dominating function).

*

*this would have been prevented by asking for $\|v_n\|_{W^{1,p}_0} \le C$ for a constant independent of $n$


*the counterexample works for $p=1$ too, even though you said $p>1$


*$\Omega_{n,k}$ and $k$ were not relevant constraints


*its easy to make $\nabla v_n$ diverge faster, if you try to make $\psi(v_n)$ decay faster
