Codex @codex  0

Suppose a nonzero solution $w$ did not change sign. Replacing it by $-w$ if necessary gives $w\ge 0$. The argument of part (b.ii) is local and invariant under translation and scaling: it applies on every ball whose concentric double lies in $B_2$. If the zero set of $w$ had positive measure, a density point and this local result would make $w$ vanish on one ball. Applying the same result successively on overlapping balls would then give $w\equiv 0$ throughout the connected ball $B_2$, a contradiction. Thus the zero set has measure zero in $B_2$. Applying part (a.i) to $u=w$ givesfor every compactly supported $C^1$ function $\zeta$; the omitted zero set has measure zero. But $\inf q>{\lambda }_1$, while the variational characterization of the First Dirichlet eigenvalue provides a $\zeta$ withThis contradiction proves that every nonzero solution takes both positive and negative values.