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

Chong Li, Shujie Li

TL;DR

The paper addresses the open problem of obtaining infinitely many distinct solutions for superlinear elliptic equations under the Ambrosetti–Rabinowitz condition without symmetry. It develops a novel invariant minimax framework built from a smooth descending flow and a characteristic mapping family, augmented by mollification of the nonlinearity to enable finite-dimensional analysis and explicit lower bounds on the generalized Morse index $M(J,u)$. The authors construct an unbounded sequence of critical points $\{u_k\}$ with $M(J,u_k)\to\infty$ and $J(u_k)\to\infty$, thereby achieving multiplicity under AR alone (without the oddness condition). The approach extends minimax theory beyond symmetry-reliant methods by combining flow-invariant mappings, smoothing, and careful PS-type limit arguments, and it applies to model nonlinearities such as $-\Delta u=|u|^{p-2}u+|u|^{q-1}$ with $2\le q<p<\frac{2N}{N-2}$ (when $N\ge3$). Overall, the work resolves a long-standing conjecture in higher dimensions and broadens the toolkit for non-symmetric, superlinear elliptic problems.

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.

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

TL;DR

The paper addresses the open problem of obtaining infinitely many distinct solutions for superlinear elliptic equations under the Ambrosetti–Rabinowitz condition without symmetry. It develops a novel invariant minimax framework built from a smooth descending flow and a characteristic mapping family, augmented by mollification of the nonlinearity to enable finite-dimensional analysis and explicit lower bounds on the generalized Morse index . The authors construct an unbounded sequence of critical points with and , thereby achieving multiplicity under AR alone (without the oddness condition). The approach extends minimax theory beyond symmetry-reliant methods by combining flow-invariant mappings, smoothing, and careful PS-type limit arguments, and it applies to model nonlinearities such as with (when ). Overall, the work resolves a long-standing conjecture in higher dimensions and broadens the toolkit for non-symmetric, superlinear elliptic problems.

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.
Paper Structure (4 sections, 11 theorems, 165 equations)

This paper contains 4 sections, 11 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Smoothness of flow for initial values
  3. The smoothing of $f\left( x,t\right)$
  4. Proofs of Theorems 1.1 and 1.2

Key Result

Theorem 1.1

Suppose $\left( f_1^{\prime }\right) \left( f_2\right) \left( f_3\right)$ hold, then the functional $J$ possesses an unbounded sequence $\left\{ u_k\right\} _{k=1}^\infty$ of critical points satisfying $\left( i\right)$$M\left( J,u_k\right) \rightarrow \infty$ as $k\rightarrow \infty$. $\left( ii\ri

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 4.1
  • Proposition 4.2
  • ...and 5 more