Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity

TL;DR

This paper proves the existence of multiple sign-changing solutions for a scalar field equation with a decaying potential and nonlinearities that are subcritical at infinity and supercritical near zero, in exterior domains.

Contribution

It establishes conditions for multiple nodal solutions in exterior domains with weak symmetry, including examples with finite symmetries in dimensions N≥4.

Findings

Existence of prescribed number of sign-changing solutions.

Conditions under which solutions exist in exterior domains.

Examples of domains with finite symmetries yielding multiple solutions.

Abstract

We consider the nonlinear elliptic equation \begin{equation*} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega), \end{equation*} in an exterior domain $\Omega$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\geq 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions.