Nonlinear Schr\"odinger equations without compatibility conditions on the potentials

TL;DR

This paper investigates the existence of solutions to nonlinear Schrödinger equations with radial potentials that do not require compatibility conditions on potential growth, extending previous results and emphasizing the role of Lebesgue spaces.

Contribution

It introduces new existence results for solutions without compatibility conditions on potentials, broadening the understanding of nonlinear Schrödinger equations with weighted potentials.

Findings

Existence of nonnegative solutions and ground states established.

Potentials with power-type estimates at zero and infinity are considered.

Results highlight the significance of Lebesgue space sums in weighted nonlinear equations.

Abstract

We study the existence of nonnegative solutions (and ground states) to the nonlinear Schr\"{o}dinger equation in $\mathbb{R}^N$ with radial potentials and super-linear or sub-linear nonlinearities. The potentials satisfy power type estimates at the origin and at infinity, but no compatibility condition is required on their growth (or decay) rates at zero and infinity. In this respect our results extend some well known results in the literature and we also believe that they can highlight the role of the sum of Lebesgue spaces in studying nonlinear equations with weights.