Existence of a positive solution to a nonlinear scalar field equation with zero mass at infinity

TL;DR

This paper proves the existence of positive solutions to a nonlinear scalar field equation with a potential that vanishes at infinity, covering cases without ground states and establishing conditions for positive bound states.

Contribution

It demonstrates the existence of positive solutions under subcritical and supercritical nonlinearities with a vanishing potential, extending previous results to new potential decay scenarios.

Findings

Existence of positive solutions under specified conditions

Solutions include cases without ground states

Conditions for positive bound states with potential decay

Abstract

We establish the existence of a positive solution to the problem $$-\Delta u+V(x)u=f(u),\qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ for $N\geq3$, when the nonlinearity $f$ is subcritical at infinity and supercritical near the origin, and the potential $V$ vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.