Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term

TL;DR

This paper investigates multiple solutions to a Caffarelli-Kohn-Nirenberg equation with an asymptotically linear term, overcoming classical conditions by employing Cerami's condition for existence results.

Contribution

It introduces a novel approach using Cerami's condition to establish multiple solutions without the Ambrosetti-Rabinowitz condition.

Findings

Established existence of multiple solutions.

Developed a new method applicable to asymptotically linear problems.

Extended the applicability of variational methods to this class of equations.

Abstract

In this paper, we study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity. In this case, the well-known Ambrosetti-Rabinowtz type condition doesn't hold, hence it is difficult to verify the classical (PS)$_c$ condition. To overcome this difficulty, we use an equivalent version of Cerami's condition, which allows the more general existence result.