Existence of nontrival $n$-harmonic maps via min-max methods

TL;DR

The paper proves the existence of nontrivial $n$-harmonic maps $u:\mathbb{S}^n\to\mathcal{N}$ for $n\ge3$ under the condition that some higher homotopy group $\pi_{n+k}({\mathcal N})$ is nonzero, by a min-max construction based on the perturbed energies $\mathcal{E}_p(u)=\int_{\mathbb S^n}|\nabla u|^p$ with $p>n$. A key technical ingredient is an $\varepsilon$-regularity result that yields fractional differentiability, enabling one-bubble compactness as $p\to n^+$ and allowing extraction of a nontrivial $n$-harmonic map either as a limit or as a bubble. The argument handles all dimensions $n\ge3$ and arbitrary smooth closed targets, providing a robust min-max framework for the critical exponent case via a one-bubble extraction and removal of singularities on $\mathbb{S}^n$. The results extend known $k=0$ theory and furnish bubbling descriptions when $k\ge1$, with implications for the structure of energy-critical maps into manifolds.

Abstract

For any $n \geq 3$ and any closed manifold $\mathcal{N}$ with $π_{n+k}(\mathcal{N}) \neq \{0\}$ for some $k \geq 0$, we establish the existence of nontrivial $n$-harmonic maps from $\mathbb{S}^n$ into $\mathcal{N}$. When $k\geq 1$, these maps naturally appear as bubbling limits of $p$-harmonic maps with $p > n$, obtained by min-max constructions in the limit $p \to n^+$.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• Corollary 2.3
• Theorem 3.1
• Lemma 3.2: Iwaniec--Sbordone a priori estimate
• proof
• Theorem 3.3
• proof : Proof of \ref{['th:epsreg']}