Let $\Omega \subset \mathbb{R}^N$, $N \geq 3$ be a bounded smooth domain. Consider the problem $$ (P) \begin{cases} -\Delta u = |u|^{p-2}u, \Omega \\ \,\,\,\,\,\,\,\,\,u = 0, \partial \Omega \end{cases} $$ where $ 2< p < 2^{*} = \frac{2N}{N-2}$. I'm trying to apply the sub-supersolution method to solve (P). For the supersolution, let $\varphi_1$ the positive eigenfunction associated to the first eigenvalue $\lambda_1$ of $-\Delta$ in $\Omega$. Now define $\overline{u} = \epsilon \varphi_1$. This is a supersolution of $(P)$ for $$ 0< \epsilon < \left(\frac{\lambda_1}{||\varphi_1||_{\infty}^{p-2}} \right)^{\frac{1}{p-2}}. $$ In fact, noting that $|\varphi_1(x)| \leq ||\varphi_1||_{\infty}$ for all $x \in \Omega$, we have \begin{align} \left(\frac{\lambda_1}{||\varphi_1||_{\infty}^{p-2}} \right)^{\frac{1}{p-2}} \leq \left(\frac{\lambda_1}{|\varphi_1(x)|^{p-2}} \right)^{\frac{1}{p-2}}. \end{align} So, \begin{align} \epsilon & \leq \left(\frac{\lambda_1}{|\varphi_1(x)|^{p-2}} \right)^{\frac{1}{p-2}} \\ \implies & 0 \leq \lambda_1 - (\epsilon \varphi_1(x))^{p-2} \\ \implies & 0 \leq \epsilon\varphi_1(x)\lambda_1 - (\epsilon \varphi_1(x))^{p-2}\epsilon\varphi_1(x)\\ \implies & 0 \leq \epsilon(-\Delta(\varphi_1)(x)) - (\epsilon \varphi_1(x))^{p-2}\epsilon\varphi_1(x) \\ \implies & 0 \leq (-\Delta(\epsilon \varphi_1)(x)) - (\epsilon \varphi_1(x))^{p-2}\epsilon\varphi_1(x) \\ \implies & 0 \leq -\Delta(\overline{u}) - |\overline{u}|^{p-2}|\overline{u}|. \end{align} Therefore, \begin{cases} -\Delta \overline{u} \geq |\overline{u}|^{p-2}\overline{u}, \Omega \\ \,\,\,\,\,\,\,\,\,\overline{u} = 0, \partial \Omega \end{cases}
Now, how can I get a subsolution $\underline{u}$ for $(P)$ such that $\underline{ u} \leq \overline{u}$ ? For the method, it is important that $\underline{u} \geq 0$ and $\underline{u} \not\equiv 0$, because the solution will verify $\underline{u} \leq u \leq \overline{u}$.
Any help is very welcome.