Positive multipeak solutions to a zero mass problem in exterior domains

Abstract

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