A nonexistence result for a nonlinear elliptic equation with singular and decaying potential

TL;DR

This paper investigates the existence of positive radial solutions to a nonlinear elliptic equation with singular and decaying potential, providing a nonexistence result for one unresolved case, thereby clarifying conditions for solution existence.

Contribution

It offers a new nonexistence result for a specific case of nonlinear elliptic equations with singular and decaying potentials, advancing understanding of solution conditions.

Findings

Established nonexistence for one of the open cases

Clarified compatibility conditions between potential decay and nonlinearity growth

Extended the theoretical understanding of solution existence criteria

Abstract

The paper deals with positive radial solutions to a nonlinear elliptic equation with singular and decaying potential, for which several existence and nonexistence results are known, resting upon suitable compatibility conditions between the decaying rate of the potential and the growth rate of the nonlinearity. The problem of the existence is still open for essentially three cases and we give a negative answer to one of such cases.