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Infinite transition solutions for an Allen-Cahn equation

Wen-Long Li

Abstract

We give another proof of a theorem of Rabinowitz and Stredulinsky obtaining infinite transition solutions for an Allen--Cahn equation. Rabinowitz and Stredulinsky have constructed infinite transition solutions as locally minimal solutions, but it is still an interesting question to establish these solutions by other method. Our result may attract the interest of constructing solutions with the shape of locally minimal solutions of Rabinowitz and Stredulinsky for problems defined on descrete group.

Infinite transition solutions for an Allen-Cahn equation

Abstract

We give another proof of a theorem of Rabinowitz and Stredulinsky obtaining infinite transition solutions for an Allen--Cahn equation. Rabinowitz and Stredulinsky have constructed infinite transition solutions as locally minimal solutions, but it is still an interesting question to establish these solutions by other method. Our result may attract the interest of constructing solutions with the shape of locally minimal solutions of Rabinowitz and Stredulinsky for problems defined on descrete group.
Paper Structure (4 sections, 16 theorems, 96 equations)

This paper contains 4 sections, 16 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['thm:6.8']}
  4. Infinite transition solutions

Key Result

Lemma 2.1

There is a constant $\bar{K}_1\geq 0$, depending on $v_0$ and $w_0$ but independent of $p\leq q\in\mathbb{Z}$ and $u\in\widehat{\Gamma}_1(v_0,w_0)$, such that $J_{1;p,q}(u)\geq -\bar{K}_1$.

Theorems & Definitions (23)

  • Definition 1.1: cf. RS
  • Lemma 2.1: RS
  • Lemma 2.2: RS
  • Lemma 2.3: RS
  • Theorem 2.4: RS
  • Lemma 2.5: RS
  • Lemma 2.6: RS
  • Lemma 2.7: RS
  • Lemma 2.8: RS
  • Lemma 2.9
  • ...and 13 more