A proof for the conjecture on superlinear problems with Ambrosetti-Rabinowitz condition

TL;DR

This paper introduces a novel minimax method to prove the existence of infinitely many solutions for superlinear elliptic equations under the Ambrosetti-Rabinowitz condition, advancing understanding in nonlinear analysis.

Contribution

It develops a new minimax approach with a characteristic mapping family invariant under flow, providing a positive solution to a long-standing open problem.

Findings

Proves existence of infinitely many solutions for superlinear elliptic equations

Introduces a new minimax approach with a characteristic mapping family

Provides a lower-bound estimate for the generalized Morse index

Abstract

This paper is devoted to exploring a new minimax approach by introducing a characteristic mapping family which is invariant under the smooth descending flow for initial value. The minimax approach is self-contained, and its features are markedly different from standard ones, as it identifies the existence of critical points and intrinsically presents a lower-bound estimate for the generalized Morse index at the corresponding critical point. This quantity can be effectively viewed as an alternative to the group action. As applications, under the Ambrosetti-Rabinowitz condition we offer a positive answer to the long-standing open problem on the existence of infinitely many distinct solutions for superlinear elliptic equations without symmetric hypothesis.