Let $\Omega \subset \mathbb{R}^{N}$ a bounded smooth domain. I would like to know if is it true that we can find a nonnegative function $\varphi \in C^{2}(\Omega)\cap C(\overline{\Omega})$ such that $$ (P)\,\,\,\begin{cases} \Delta \varphi \geq 0, \,\,\,\Omega\\ \,\,\,\,\varphi = 0, \partial \Omega. \end{cases} $$
The context: I'm trying to find a non trivial nonnegative subolution for the problem \begin{cases} -\Delta u = \lambda u + |u|^{p-2}u, \Omega \\ \,\,\,\,\,\,\,\,\,u = 0,\,\,\,\,\,\,\,\,\,\,\,\partial \Omega, \end{cases} where $0 < \lambda < \lambda_1$ and $2 < p $ [here $\lambda_1$ is the first laplacian eigenvalue]. That is, an function $\underline{u} \geq 0$, not null, such that \begin{cases} -\Delta \underline{u} \leq \lambda \underline{u} + |\underline{u}|^{p-2}\underline{u}, \Omega \\ \,\,\,\,\,\,\,\,\,\,\underline{u} = 0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial\Omega. \end{cases} If I would find a function $\varphi \geq 0$ verifying (P), so I would get $$ -\Delta \varphi \leq 0 \leq |\varphi|^{p-2} \varphi, $$ which implies, $$ -\Delta \varphi - \lambda \varphi \leq -\Delta \varphi \leq |\varphi|^{p-2} \varphi, $$ so a get what I need: $$-\Delta \varphi \leq \lambda \varphi + |\varphi|^{p-2} \varphi.$$
Any help, reference or tip is very welcome.
No, there is no such $\varphi$ except the trivial solution. Since $-\Delta \varphi \leqslant 0$ in $\Omega$ and $\varphi =0$ on $\partial \Omega$, the maximum principle implies that $\varphi \leqslant 0$ in $\Omega$. Thus, the only nonnegative $\varphi$ which satisfies this is $\varphi \equiv 0$ in $\Omega$.