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Nodal solutions to Paneitz-type equations

Jurgen Julio-Batalla, Jimmy Petean

Abstract

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $Δ^2 u -αΔu +βu = u^q$, where $α$ and $β$ are positive constants satisfying that $α^2 \geq 4 β$. We let ${\bf m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f = \frac{n-{\bf m}+4}{n-{\bf m}-4}$, $q_f = \infty$ if $n\leq {\bf m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.

Nodal solutions to Paneitz-type equations

Abstract

On a closed Riemannian manifold with a proper isoparametric function we consider the equation , where and are positive constants satisfying that . We let be the minimum of the dimensions of the focal varieties of and , if . We prove the existence of infinitely many nodal solutions of the equation assuming that . The solutions are -invariant. To obtain the result, first we prove a estimate for positive -invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.
Paper Structure (3 sections, 6 theorems, 67 equations)

This paper contains 3 sections, 6 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Isoparametric functions and compactness
  3. Mountain pass critical points

Key Result

Theorem 1.1

Let $f$ be a proper isoparametric function on $(M,g)$. Assume that $1<q<q_f$. Then for any $0 <a < b$, the set $S=S_{a , b} = \{(u,\alpha , \beta ) : u$ is a positive , $f$-invariant solution of Equation (Paneitz) in $C^{4,\gamma}(M)$ and $\alpha , \beta \in [a,b]$ , with $\alpha^2 \geq 4\beta \}$

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 1 more