Let $\Omega \subset \mathbb{R}^N$ be a bounded domain, $g \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})$ and $g$ bounded. If $G(x,u) := \int_0^u g(x,s) ds$, show that
$$\lim\limits_{|s| \rightarrow \infty} G(x,s) = \infty \ \text{uniformly on} \ \ \Omega \Longrightarrow \int_{\Omega} G(x,v(x)) dx \rightarrow \infty \ \text{when} \ ||v|| \rightarrow \infty \ \text{and} \ v \in \ \text{ker} \ (\Delta + \lambda_j Id).$$
This $g$ arises from the problem
$$\begin{align*} \begin{cases} - \Delta u &= \lambda u + g(x,u), x \in \Omega,\\ u &= 0, x \in \partial \Omega, \end{cases} \end{align*}$$
where $\lambda = \lambda_j$, $\lambda_{j-1} < \lambda_j = \lambda_{j+1} = \ldots = \lambda_{j+m-1} < \lambda_{j+m}$ and $\{ \varphi_n \}$ are the eigenfunctions associated to $\lambda_n$ for the eigenvalue problem of the operator $-\Delta$.
$\textbf{Hint:}$ consider the projection of $\text{ker} \ (\Delta + \lambda_j Id) \backslash \{ 0 \}$ into the unit sphere and use the fact that the dimension of this space is finite.
I can't see how the hint helps me to prove the implication above, so I tried this:
As $g$ is bounded, follows from the hypothesis that
$$\lim\limits_{||v|| \rightarrow \infty} \int_{\Omega} G(x,v(x)) dx \geq \lim\limits_{||v|| \rightarrow \infty} - M |v|_{L^1(\Omega)} \geq \lim\limits_{||v|| \rightarrow \infty} - M c ||v|| = - \infty,$$
where $c$ denotes the constant of the immersion $H_0^1(\Omega) \hookrightarrow L^1(\Omega)$.
I would like to know how the hint helps me to solve this problem, because all I can see with this hint is the unit sphere of $\text{ker} \ (\Delta + \lambda_j Id)$ is compact.
Thanks in advance!
$\textbf{P.S.:}$ sorry for the bad title for this topic, but I couldn't think in a better title.