Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems

Abdelrazek Dieb, Isabella Ianni, Alberto Saldaña

Abstract

We prove the uniqueness and nondegeneracy of least-energy solutions of a fractional Dirichlet semilinear problem in sufficiently large balls and in more general symmetric domains. Our proofs rely on uniform estimates on growing domains, on the uniqueness and nondegeneracy of the ground state of the problem in RN , and on a new symmetry characterization of the eigenfunctions of the linearized eigenvalue problem in domains which are convex in the x1 - direction and symmetric with respect to a hyperplane reflection.

Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems

Abstract

We prove the uniqueness and nondegeneracy of least-energy solutions of a fractional Dirichlet semilinear problem in sufficiently large balls and in more general symmetric domains. Our proofs rely on uniform estimates on growing domains, on the uniqueness and nondegeneracy of the ground state of the problem in RN , and on a new symmetry characterization of the eigenfunctions of the linearized eigenvalue problem in domains which are convex in the x1 - direction and symmetric with respect to a hyperplane reflection.
Paper Structure (12 sections, 19 theorems, 131 equations)

This paper contains 12 sections, 19 theorems, 131 equations.

Table of Contents

  1. Introduction and main results
  2. Notation and preliminary results
  3. The linearized problem
  4. Least-energy solutions
  5. Ground-state solutions in $\mathbb R^N$
  6. A symmetry result for the linearized problem
  7. Second eigenfunctions: a variational characterization
  8. Polarization of second eigenfunctions in symmetric domains
  9. The proof of Theorem \ref{['sym']}
  10. Uniqueness and nondegeneracy of least-energy solutions in large symmetric domains
  11. Convergence
  12. Nondegeneracy and uniqueness

Key Result

Theorem 1.1

Let $N\geq 2$ and let $\Omega=B_R\subset\mathbb R^N$ be a ball of radius $R>0$. Let $s\in (0,1)$, $p\in(1,2^\star_s-1),$ and $\lambda> 0$. There exists $R_0:=R_0(p,\lambda, s)\geq 1$ such that, if $R\geq R_0$, then problem Peps has a unique least-energy solution and it is nondegenerate.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 35 more