# Could this integral be estimated with a positive constant?

Let $$\Omega$$ be an opend bounded subset of $$\mathbb{R}^n$$ and let $$p, q$$ be two real numbers such that $$p, q\geq 1$$. Let $$(w_n)_n\subset W_0^{1, p}(\Omega)$$ and $$(z_n)_n\subset W_0^{1, q}(\Omega)$$ such that $$\exists w\in W_0^{1, p}(\Omega)$$ such that $$w_n\longrightarrow w \quad \mbox{ in } L^{r}(\Omega) \quad \mbox{ for } \ 1\leq r < p^{\ast}$$ and $$w_n\longrightarrow w \quad \mbox{ a.e. in } \Omega.$$ Moreover, fix $$k\geq 1$$ and consider $$\Omega_{n, k}:=\left\lbrace x\in\Omega \mid \vert w_n(x), z_n(x)\vert > k\right\rbrace.$$ I would like to show that the integral $$\int_{\Omega\setminus\Omega_{n, k}} F(x, w_n, z_n) w \vert\nabla z_n\vert^{q} dx$$ can be estimated with a positive constant, i.e. $$\exists c\in\mathbb{R}$$ such that $$\displaystyle\int_{\Omega\setminus\Omega_{n, k}} F(x, w_n, z_n) w \vert\nabla z_n\vert^{q} dx\leq c$$.

Here, I assume $$F:\Omega\times\Omega\times\mathbb{R}\to\mathbb{R}$$ such that $$\sup_{\vert (u, v)\vert\leq t} \vert F(\cdot, u, v)\vert\in L^{\infty}(\Omega)$$ for any $$t>0$$.

• Any assumptions on the sequence $z_n$? Jun 27 '20 at 16:47
Since $$z_n \in W_{0}^{1, q}$$, your only hope on estimating the integral of $$|\nabla z_n|^q$$ is to put it in $$L^1$$ and pull out an $$L^\infty$$ norm of the $$F$$ term: $$\int_{\Omega \setminus \Omega_{n,k}} |F(x, w_n, z_n)||w||\nabla z_n|^q \, dx \leq \|{z_n}\|_{W^{1 , q}}\sup_{x \in \Omega \setminus \Omega_{n, k}} |F(x, w_n(x), z_n(x))||w(x)|.$$ So to get a uniform bound we'd first need to know that the $$z_n$$'s are uniformly bounded in $$W^{1, q}$$, or at least that their gradients are bounded uniformly in $$L^q$$. Meanwhile the supremum term is finite for each $$k$$, since $$\Omega_{n,k}$$ is the set where $$w_n, z_n$$ are greater than $$k$$, but this bound potentially depends on $$k$$. Therefore we'd probably also need some assumptions on $$F$$'s $$L^\infty$$ behavior.
If there are no such assumptions on $$F$$ or $$z_n$$, we can cook up a counterexample. Let $$F \equiv 1$$ and let $$w_n \equiv w \in W_0^{1, q}$$ be some smooth bump function. Then let $$z_n$$ be some sequence in $$W_0^{1, q}$$ such that $$\sup_{x \in \Omega} |z_n| \leq C$$ for all $$n$$, but such that $$\|\nabla z_n\|_{L^q} \to \infty$$ as $$n \to \infty$$ (for instance, on $$\mathbb{R}$$ we could let $$z_n$$ be something like $$\sin(nx)$$, perhaps a polygonal version of this). Then the integral above is always over all of $$\Omega$$ for $$k > C$$ and in fact it's equal to $$\int_{\Omega} |w(x)||\nabla z_n|^q\, dx.$$ So as long as we choose, say, $$w$$ to be equal to 1 on some set of large enough measure, this integral will go to infinity as $$n \to \infty$$.