Let $h:[0,\infty)\to\mathbb{R}$ be a continuous nondecreasing function and $\gamma$ is a positive continuous function over $\Omega$ such that $0<\gamma(x)<1$ for every $x\in\Omega$, which is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 2$. Let $W_0^{1,p}(\Omega)$ be the Sobolev space with zero boundary values in the trace sense. Define the functional $I:W_0^{1,p}(\Omega)\to\mathbb{R}$ by $$ I(u)=\int_{\Omega}H(u)\,\mathrm dx, $$ where $$ H(t)= \begin{cases} \int_{0}^{t}h(s)\,s^{-\gamma(x)}\,\mathrm ds,\quad \text{if} \quad t\geq 0,\\ 0\quad\text{ if }t<0. \end{cases} $$ Then for every $\phi\in C_c^{1}(\Omega)$, is it true that $$ \langle I'(u),\phi\rangle=\int_{\Omega}h(u)\,u^{-\gamma(x)}\phi\,\mathrm dx, $$ where $I^{'}$ is the Gateaux derivative of $I$.
I tried in the following way. Since $h$ is continuous, we have $H^{'}(t)=h(t)t^{-\gamma(x)}$ for every $t\geq 0$ and $H^{'}(t)=0$ for every $t<0$. Then $$ \langle I^{'}(u),\phi\rangle = \lim_{t\to 0}\frac{I(u+t\phi)-I(u)}{t} \\=\lim_{t\to 0}\frac{\int_{\Omega}\{H(u+t\phi)-H(u)\}\,\mathrm dx}{t}=\lim_{t\to 0}\frac{\int_{\Omega}\{t\phi H^{'}(u+t\theta\phi)\}\,\mathrm dx}{t} $$ for some $\theta\in(0,1)$. After this step, I am unable to proceed. Because $u+t\phi$ may have different sign. Can someone please help on how to proceed from here. Thanks.