# Continuous map from $L^{p+1}(\Omega)$ to $L^\frac{p+1}{p}(\Omega)$

I am having trouble showing the following map is continuous:

Let $$n\geq2$$, $$\Omega\subset \mathbb{R}^n$$ be a bounded domain, and $$1, where $$2^*=\frac{2n}{n-2}$$ is the Sobolev conjugate of $$2$$. The map $$\tau:L^{p+1}(\Omega,\mathbb{R})\rightarrow L^\frac{p+1}{p} (\Omega,\mathbb{R})$$ is given by $$u\mapsto |u|^{p-1}u.$$

I could show $$\tau$$ is bounded, however, I could not show it is continuous. Could someone tell me how I should approach this?

Thanks!

• I would suggest approximation by smooth functions and difference = integral of derivative? Commented Jul 28 at 16:43

This operator is a specific variant of the Nemytskii Operator.

Here, one can think about using brute force (i.e. dominated convergence).

Let $$u\in L^{p+1}(\Omega, \mathbb{R})$$ and $$(u_k)_{k \in \mathbb{N}} \subseteq L^{p+1}(\Omega, \mathbb{R})$$ be a sequence that converges to $$u$$ in $$L^{p+1}(\Omega, \mathbb{R})$$. Then, every subsequence of $$u_k$$ has a subsequence (that we denote by $$u_{n_k}$$) that converges to $$u$$ pointwise a.e.. Most importantly, $$\lvert u_{n_k} \rvert^{p-1}u_{n_k} \rightarrow \lvert u \rvert^{p-1}u$$ pointwise a.e..

Moreover for all $$k \in \mathbb{N}$$: $$\big\lvert \lvert u_k \rvert^{p-1}u_k - \lvert u \rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}} ~\mathrm{d}x \leq 2^{\tfrac{1}{p}}\left( \big \lvert \lvert u_k\rvert^{p-1}u_k \big \rvert^{\tfrac{p+1}{p}}+ \big \lvert \lvert u\rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}} \right)$$ Note that $$2^{\tfrac{1}{p}}\left( \big \lvert \lvert u_{n_k}\rvert^{p-1}u_{n_k} \big \rvert^{\tfrac{p+1}{p}}+ \big \lvert \lvert u\rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}} \right) \rightarrow 2^{\tfrac{1}{p}}\left( \big \lvert \lvert u\rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}}+ \big \lvert \lvert u\rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}} \right)$$ in $$L^1(\Omega, \mathbb{R})$$ and pointwise a.e. due to the assumed convergence of $$u_k$$ in $$L^{p+1}(\Omega, \mathbb{R})$$. Then, the generalized dominated convergence theorem (Evans, Gariepy - Measure Theory and fine properties of functions, Theorem 1.20) yields $$\int_{\Omega}\big\lvert \lvert u_{n_k} \rvert^{p-1}u_{n_k} - \lvert u \rvert^{p-1}u \big \rvert^{\tfrac{p+1}{p}} ~\mathrm{d}x \rightarrow 0.$$

As the subsequence from which $$u_{n_k}$$ has been extracted was arbitrary, the above limit also holds for the general sequence $$u_k$$ which is your desired continuity.

• When extracting the pointwise a.e. convergence, one can also directly obtain a corresponding dominating function (see Brezis book). This would simplify the proof.
– daw
Commented Aug 1 at 12:42
• Thank you for the nice hint, I‘ll later take a look at the book! Commented Aug 1 at 14:01