How to show that $J(u) = \frac1p \int_{\Bbb{R}^N} |u|^{p} \ dx$ is $C^1$ in $W^{1, p}(\Bbb{R}^N)$?

Question

How to show that $J(u) = \frac1p \int_{\Bbb{R}^N} |u|^{p} \ dx$ is $C^1$ in $W^{1, p}(\Bbb{R}^N)$? Assume $N \geq 3$ and $2 \leq p < N$.

I computed the Gateaux-derivative $$ J'_G(u)v = \int_{\Bbb{R}^N} |u|^{p - 2}uv \ dx, \quad v \in W^{1, p}(\Bbb{R}^N), $$ and now I am trying to show that it is continuous.

Let $u_n \to u$ in $W^{1, p}(\Bbb{R}^N)$. By the continuous Sobolev embedding, $u_n \to u$ in $L^{p^*}(\Bbb{R}^N)$. By Vainberg's theorem there is a subsequence and $g \in L^{p^*}(\Bbb{R}^N)$ such that $$ u_n(x) \to u(x), \quad |u_n(x)| \leq g(x) \quad \text{ a.e. in } \Bbb{R}^N. $$ Of course, by the first condition, $$ |u_n(x)|^{p - 2}u_n(x) \to |u(x)|^{p - 2}u(x) \quad \text{ a.e. in } \Bbb{R}^N. $$ Also, $$ ||u_n|^{p - 2}u_n - |u|^{p - 2}u|^{\frac{p^*}{p - 1}} \leq C|u_n|^{p^*} + C|u|^{p^*} \leq 2C g^{p^*} \in L^1 (\Bbb{R}^N). $$ Now, I would like to use Hölder's inequality to bound $(J_G'(u_n) - J'_G(u))v$ by the norm $L^{p^*}$ of $v$ and the norm $L^{p^*/(p-1)}$ of $|u_n|^{p - 2}u_n - |u|^{p - 2}u$, and then use the Dominated Convergence Theorem, but it seems impossible.

How to proceed?

Thanks in advance and kind regards.

Answer 1

Consider the inequality $||x|^{p-2}x-|y|^{p-2}y| \leq C|x-y|(|x|+|y|)^{p-2}$, which is valid for $p>2$, and $(u_n)_{n}\subset W^{1,p}(\mathbb{R^n})$ such that $||u_n-u||_{W^{1,p}} \to 0$ when $n \to \infty$. \begin{align*} |J'_G(u)(v)-J'_G(u_n)(v)|& \leq \int_{\mathbb{R^n}}||u|^{p-2}u-|u_h|^{p-2}u_n||v|dx \\ &\leq C \int_{\mathbb{R^n}} |u-u_n|(|u|+|u_n|)^{p-2}|v| dx \\ & \leq C ||u-u_n||_{Lp}\left (\int_{\mathbb{R^n}} (|u|+|u_n|)^{(p-2)p/(p-1)}|v|^{p/(p-1)}|dx \right)^{(p-1)/p}. \\ \end{align*} Observe that $\frac{1}{p-1}+\frac{p-2}{p-1}=1$, by Hölder inequality again \begin{align*} \int_{\mathbb{R^n}} (|u|+|u_n|)^{(p-2)p/(p-1)}|v|^{p/(p-1)}|dx &\leq \left(\int_{\mathbb{R}^n}(|u|+|u_n|)^{p}dx \right)^{(p-2)/(p-1)} \left(\int_{\mathbb{R^n}} |v|^pdx \right)^{1/(p-1)} \end{align*} And $(|u|+|u_n|)^p \leq 2^p \max\{|u_n|^p,|u|^p\} \leq 2^p(|u|^p+|u_n|^p)$, then \begin{align*} \int_{\mathbb{R^n}} (|u|+|u_n|)^{(p-2)p/(p-1)}|v|^{p/(p-1)}|dx &\leq \left(2^p\int_{\mathbb{R}^n}(|u|^p+|u_n|^p)dx \right)^{(p-2)/(p-1)} ||v||_{L_p}^{p/(p-1)} \\ & \leq 2^{p(p-2)/(p-1)}(||u||_{L_p}^p+||u_n||_{L_p}^p)^{(p-2)/(p-1)}||v||_{L_p}^{p/(p-1)} \end{align*} Finally \begin{align*} |J'_G(u)(v)-J'_G(u_n)(v)| & \leq C||u-u_n||_{Lp}\cdot 2^{p-2}(||u||_{L_p}^p+||u_n||_{L_p}^p)^{(p-2)/p} ||v||_{L_p}\xrightarrow[n \to \infty ]{} 0 \end{align*}

From this the continuity of $J'_G(u)$ follows.