I want to know if the following problem
$$\begin{align*} \begin{cases} -\Delta u &= u \ \text{in} \ \Omega,\\ u &= 0 \ \text{on} \ \partial \Omega, \end{cases} \end{align*}$$ where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary, has a non-trivial weak solution.
$\textbf{My attempt:}$
Observe that if there is a non-trivial weak solution $u \in H_0^1(\Omega)$, then $1$ is an eigenvalue of $-\Delta$.
Consider two cases.
Case $1$) $\lambda_1 < 1$:
From the weak formulation of the eigenvalue problem of $-\Delta$,
$$\int_{\Omega} \nabla u \nabla v dx - \int_{\Omega} u v dx = 0, \forall v \in H_0^1(\Omega).$$
If $v = \varphi_1$, then
$$\int_{\Omega} \nabla u \nabla \varphi_1 dx - \int_{\Omega} u \varphi_1 dx = 0. (1)$$
From the weak formulation of the eigenvalue problem of $-\Delta$,
$$\int_{\Omega} \nabla \varphi_1 \nabla v dx - \lambda_1 \int_{\Omega} \varphi_1 v dx = 0, \forall v \in H_0^1(\Omega).$$
If $v = u$, then
$$\int_{\Omega} \nabla \varphi_1 \nabla u dx - \lambda_1 \int_{\Omega} \varphi_1 u dx = 0. (2)$$
From $(1)$, $(2)$ and the hypothesis that $\lambda_1 < 1$, follows that
$$0 = \int_{\Omega} \nabla \varphi_1 \nabla u dx - \lambda_1 \int_{\Omega} \varphi_1 u dx > \int_{\Omega} \nabla u \nabla \varphi_1 dx - \int_{\Omega} u \varphi_1 dx = 0,$$
which is an absurd.
Case $2$) $\lambda_1 = 1$:
From the weak formulation of the eigenvalue problem of $-\Delta$,
$$\int_{\Omega} \nabla u \nabla v dx - \int_{\Omega} u v dx = 0, \forall v \in H_0^1(\Omega).$$
I'm stuck here and I don't know how to proceed. I would like to know what I can conclude in the case $2$.
Thanks in advance!
