Let $\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$ be an energy functional for bounded domain $\Omega$ such that $\phi'$ be its Frechet derivative, and there exists a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}}$ bounded in $H_{0}^{1}(\Omega)$ such that $\phi(u_{n})\to d$ and $\phi'(u_{n})\to 0$ as $n\to\infty$. Assume that $u_{n}\to u$ weakly in $H_{0}^{1}(\Omega)$ and define $\langle\,\cdot\, , \,\cdot\,\rangle$ as the inner dot product in $H_{0}^{1}(\Omega)$.

Claim : $\langle\phi'(u_{n})-\phi'(u),u_{n}-u\rangle\to 0$ as $n\to\infty$.

My question is how to show that claim. My attempt so far is to separate it into : $$\langle\phi'(u_{n}),u_{n}-u\rangle - \langle\phi'(u),u_{n}-u\rangle$$ The second term tends to zero as $n\to\infty$ by weak convergence of $\{u_{n}\}_{n\in\mathbb{N}}$ but I am not sure how to show that the first term tends to zero since $u_{n}-u$ still depends on $n$ so I do not know whether I could use the assumption that $\phi'(u_{n})\to0$ as $n\to\infty$.

Any help is much appreciated! Thank you very much!


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