Let $u \in L^{\infty}(\Omega)\cap H_{0}^{1}(\Omega)$ and define a cut-off function $\eta_{R} \in C_{0}^{\infty}(\mathbb{R})$ for $\Omega \subset \mathbb{R}$ an unbounded (interval) domain as follows

$$\eta_{R}(x):=\begin{cases} 1 &, \text{if }|x|\leq R\\ 0 &, \text{if } x\geq R+1\text{ or }x\leq-R-1\\ 0<\eta_{R}(x)<1 &, \text{ other }x \end{cases} $$

Now define a function $u_R = u\eta_{R}$. Define a functional $I[u] = ||u||_{H_{0}^{1}(\Omega)}^{2} - ||u||_{L^{p}(\Omega)}^{p}$ for $2\leq p <\infty$.

I would like to show that $I[u_{R}] = I[u]+o(1)$ as $R\to\infty$. This is my attempt so far:

\begin{align*} |I[u]-I[u_{R}]| &\leq |\, ||u||_{H_{0}^{1}(\Omega)}^{2}-||u_{R}||_{H_{0}^{1}(\Omega)}^{2}\,| + |\, ||u||_{L^{p}(\Omega)}^{p} - ||u_{R}||_{L^{p}(\Omega)}^{p}\,| \\ &\leq |\, ||u||_{L^{2}(\Omega)}^{2} - ||u||_{L^{2}(\Omega)}^{2}\,| + |\, ||\nabla u||_{L^{2}(\Omega)}^{2}- ||\nabla u_{R}||_{L^{2}(\Omega)}^{2}\,|+ |\, ||u||_{L^{p}(\Omega)}^{p} - ||u_{R}||_{L^{p}(\Omega)}^{p}\,|\\ &= A + B + C \end{align*}

So, I would like to see the estimate one by one. First, I will start from $A$. Observe that \begin{align*} A &\leq \int_{\Omega}|u^{2}\eta_{R}^{2}-u^{2}|dx \\ &\leq \sup\limits_{\Omega}u^{2}\int_{\Omega}|1-\eta_{R}^{2}|dx\\ &\leq 2\sup\limits_{\Omega}u^{2}\int_{\Omega}|1-\eta_{R}|dx \end{align*}

Similarly for $C$, I will obtain the following estimate $$C \leq p\sup\limits_{\Omega}|u|^{p}\int_{\Omega}|1-\eta_{R}|dx$$

Finally, for $B$, I would obtain $$B \leq \int_{\Omega}|\partial_{x}(u\eta_{R})|^{2} - (\partial_{x}u)^{2}|dx$$

So, I have two main problems here.
1. How to ensure that $\int_{\Omega}|1-\eta_{R}|dx \to 0$ as $R\to\infty$ rigorously?
2. What should I do to estimate $B$ since I am not sure what to do with the term $\partial_{x}\eta_{R}$ here.

Any help will much be appreciated!


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