Suppose that $(w_n)$ is a sequence of sign-changing functions such that $w_n \to \varphi_1$ uniformly on the bounded smooth set $\Omega \subset \mathbb R^N$, and $w_n = 0$ on $\partial \Omega$. Is it true that $\varphi_1$ must change sign?
Motivation: in my book, $\varphi_1$ is the first eigenvalue of the Laplacian, and the authors claim that the convergence would imply a contradiction as $\varphi_1$ doesn't change sign. What I'm thinking is that the minimum of $w_n$ (which is negative) could be closer and closer to $0$, with the minimum point tending to the boundary, such that the uniform convergence still holds and therefore there is no contradiction. What am I missing?
Thanks in advance.
EDIT
More details on $w_n$: it solves $$ - \Delta w_n = g_n w_n + \lambda w_n \text{ in } \Omega, \qquad w_n = 0 \text{ on } \partial \Omega $$ with $g_n \to \lambda_1 - \lambda$, $\lambda < \lambda_1$, $\lambda_1$ the first eigenvalue of the Laplacian