A Linear Perturbation Producing a Sequentially Strict Maximum under a Coercive Bound

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, let $A$ be a nonempty subset of $H$, and let $\Phi:A\to\mathbb{R}$ have closed superlevel sets in $H$. Suppose there are $C\in\mathbb{R}$ and a positive $\kappa\in\mathbb{R}$ such that

$$\Phi(x)\le C-\kappa\,|x|^{2}\qquad\text{for every }x\in A.$$

Then for every positive $\gamma\in\mathbb{R}$ there exist $p\in H$ with $|p|\le\gamma$ and $\bar{x}\in A$ such that the function $A\to\mathbb{R}$ whose value at $x\in A$ is

$$\Phi(x)-\langle p,x\rangle$$

attains a sequentially strict maximum on $A$ at $\bar{x}$.