Let $\Omega$ be a bounded smooth domain in $R^n$, $n \geq 2.$
$f:\overline{\Omega} \times R \rightarrow R$ is a continuous function of both variables, and satisfies the growth condition
i.e. $\vert f \left( x,s\right) \vert \leq c\vert s\vert ^{p-1} +b\left( x \right) $
where $c>0$ is a constant, $b\left(x\right) \in L^{p'}\left(\Omega \right)$ with $\frac{1}{p}+\frac{1}{p'}=1,$ and $1 \leq p < \infty$ if $n=2,$ or $1\leq p \leq \frac{2n}{n-2}$ if $n \geq 3.$
We assume that $\quad \quad \mathop {\lim \inf }\limits_{s \to +\infty } \frac{f\left(x,s\right)}{s} > \lambda_1$ uniformly in $\overline{\Omega} \quad (\star)$
where $\lambda_1$ is the first eigenvalue of $\left(−\Delta, H_0^1 \right).$
Show that there are constants $\mu > \lambda_1$ and $c$ such that $f\left(x, s\right) \geq \mu s − c,$ for all $s > 0.$
I think that we only need to use the condition $(\star).$ Could anyone help me to give a solution for the problem! thank you!