Positive multipeak solutions to a zero mass problem in exterior domains

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass $$-\mathrm{\Delta }u=f(u),u\in D_0^{1,2}({\mathrm{\Omega }}_R),$$ where ${\mathrm{\Omega }}_R:=\{x\in {\mathbb{R}}^N:\mathrm{\mid }u\mathrm{\mid }>R\}$ with $R>0$, $N\ge 4$, 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 $R\to \mathrm{\infty }$.