Return to Answer

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\\ J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\\ dx\\ \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \\ dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest A Primer of Nonlinear Analysis by Antonio Ambrosetti and Giovanni Prodi (resp. my mathematical father and grandfather ;-).

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\, J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\, dx\, \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \, dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest A Primer of Nonlinear Analysis by Antonio Ambrosetti and Giovanni Prodi (resp. my mathematical father and grandfather ;-).

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\\ J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\\ dx\\ \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \\ dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest A Primer of Nonlinear Analysis by Antonio Ambrosetti and Giovanni Prodi resp. my mathematical father and grandfather ;-).

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\\ J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\\ dx\\ \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \\ dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest A Primer of Nonlinear Analysis by Antonio Ambrosetti and Giovanni Prodi (resp. my mathematical father and grandfather ;-).

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\\ J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\\ dx\\ \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \\ dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c.

Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get $$\bigg|\\ J(u+v)-J(u)-(p+1)\int_\Omega |u|^p v\\ dx\\ \bigg| \le p(p+1)\int_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \\ dx.$$ Use the Hoelder inequality to check that the RHS is actually $o(\|v\|_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^*$, otherwise $J(u)$ may be not even defined for some $u$).

For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest A Primer of Nonlinear Analysis by Antonio Ambrosetti and Giovanni Prodi resp. my mathematical father and grandfather ;-).