Positive multipeak solutions to a zero mass problem in exterior domains

TL;DR

This paper proves the existence of multiple positive solutions with multiple peaks for a nonlinear scalar field equation in exterior domains, showing the number of solutions grows as the domain expands.

Contribution

It introduces new methods to establish the existence of arbitrarily many positive multipeak solutions in exterior domains for a zero mass problem.

Findings

Number of solutions increases with domain size R

Existence of multipeak solutions in exterior domains

Solutions are positive and have multiple peaks

Abstract

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 $R \to \infty$.