  We establish the existence of positive multipeak solutions to the nonlinear
scalar field equation with zero mass $$-\Delta u = f(u), \qquad u\in
D_0^{1,2}(\Omega_R),$$ where $\Omega_R:=\{x \in \mathbb{R}^N:|u|>R\}$ with
$R>0$, $N\geq4$, and the nonlinearity $f$ is subcritical at infinity and
supercritical near the origin. We show that the number of positive multipeak
solutions becomes arbitrarily large as $R \to \infty$.
