We study the problem \(\begin{aligned} {\left\{ \begin{array}{ll} {}-\varepsilon ^2 \mathscr {M}_{\lambda , \Lambda }^{\pm } (D^2 u) =f(u) & \text {in}\;\;\Omega ,\\ \,u=0 & \text {on}\;\;\partial \Omega , \end{array}\right. } \end{aligned}\) where f is a nonnegative, locally Lipschitz continuous function, \(\varepsilon \) is a positive parameter, and \(\Omega \) is a smooth bounded domain of \(\mathbb {R}^N\) . We aim to show conditions under which for \(\varepsilon \) close enough to zero there exist at least two positive solutions \(u_\varepsilon <v_\varepsilon \) , verifying \(\Vert u_\varepsilon \Vert _\infty<1< \Vert v_\varepsilon \Vert _\infty \) and \(u_\varepsilon , v_\varepsilon \rightarrow 1\) uniformly on compact subsets of \(\Omega \) as \(\varepsilon \rightarrow 0\) , provided f has a positive zero. The hypotheses utilized involve critical exponents, and in this context, a new Liouville-type theorem is presented.
