Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball

Francesca Dalbono, Matteo Franca, Andrea Sfecci

Abstract

In this paper we show that the number of radial positive solutions of the following critical problem $$ Δ_p u(x) + λK(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q= \frac{np}{n-p}$, $\frac{2n}{n+2} \le p \le 2$ and $x \in \mathbb{R}^n$, undergoes a bifurcation phenomenon. Namely, the problem admits one solution for any $λ>0$ if $K$ is steep enough at $0$, while it admits no solutions for $λ$ small and two solutions for $λ$ large if $K$ is too flat at $0$. The existence of the second solution is new, even in the classical Laplace case. The proofs use Fowler transformation and dynamical systems tools.

A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball

Abstract

In this paper we show that the number of radial positive solutions of the following critical problem where , and , undergoes a bifurcation phenomenon. Namely, the problem admits one solution for any if is steep enough at , while it admits no solutions for small and two solutions for large if is too flat at . The existence of the second solution is new, even in the classical Laplace case. The proofs use Fowler transformation and dynamical systems tools.

Paper Structure

This paper contains 9 sections, 39 theorems, 152 equations, 5 figures.

Table of Contents

  1. Introduction
  2. On the proofs of the theorems
  3. Fowler transformation, basic notation and preliminaries
  4. Basic properties of the function $\boldsymbol{R(d)}$.
  5. The asymptotic estimate of $\boldsymbol{R(d)}$ for $\boldsymbol{d}$ large.
  6. The case $\boldsymbol{\ell < \ell_{p}^*}$.
  7. The case $\boldsymbol{\ell \geq \ell_{p}^*}$.
  8. Proof of Lemma \ref{['l.new.technical']} and Proposition \ref{['p.parzialebis']}.
  9. Proof of the theorems

Key Result

Theorem 1

Assume that ${\mathcal{K}}$ satisfies $\boldsymbol{({\rm H}_\ell)}$.

Figures (5)

  • Figure 1: A sketch of the graph of the function $R(d)$, when $\boldsymbol{({\rm H}_\ell)}$ and $\boldsymbol{({\rm H}_\uparrow\!\!\!)}$ are assumed. In the critical case $\ell=\ell^*_{p}$, we have three possible alternatives.
  • Figure 2: A sketch of the graph of the function $R(d)$, in the setting of Theorem \ref{['t.main.tre']}. In the critical case $\ell=\ell^*_{p}$, we have three possible alternatives.
  • Figure 3: On the left, the energy levels of ${\mathcal{H}}$ at fixed time $\tau$, with the homoclinic orbit $\boldsymbol{\Gamma_{\tau}}$. On the right, the position of the set $\boldsymbol{ \Gamma_{\tau_2}}$ with respect to the set $\boldsymbol{\Gamma_{\tau_1}}$ in the case $K(\tau_2)>K(\tau_1)$.
  • Figure 4: The position of the points $\boldsymbol{Q}(\tau,a)$, $\boldsymbol{R}(\tau,a)$, and $\boldsymbol{R}^1(\tau,a)$.
  • Figure 5: The constructions needed in the proof of Lemma \ref{['l.new.technical']}

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Remark 7
  • Remark 8
  • Lemma 9
  • proof
  • ...and 69 more