Pohozaev manifold constraint for solving nonlinear Schr\"odinger equations with potentials vanishing at infinity

TL;DR

This paper establishes the existence of positive solutions for a class of nonlinear Schrödinger equations with decaying potentials using a novel approach involving soliton compositions and Pohozaev manifold projections.

Contribution

It introduces a new method combining translated and dilated solitons with Pohozaev manifold projections to prove existence results for Schrödinger equations with vanishing potentials.

Findings

Existence of positive solutions under decay conditions on the potential.

Development of a new composition technique for solitons.

Application of Pohozaev manifold constraints to nonlinear Schrödinger equations.

Abstract

Existence of a positive solution for a class of nonlinear Schr\"odinger equations with potentials which decay to zero at infinity, with an appropriate rate, approaching zero mass type limit scalar field equations, is established via a new composition of two translated and dilated solitons and its projection on the so called Pohozaev manifold.