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A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

Giovanni Molica Bisci, Paolo Malanchini, Simone Secchi

Abstract

We consider the boundary value problem $$ \cases{ -Δ_γu = λu + \left\vert u \right\vert^{2^*_γ-2}u &in $Ω$\cr u = 0 &on $\partialΩ$,\cr } $$ where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0). $$ We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei.

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

Abstract

We consider the boundary value problem where is an open bounded domain in , , while is the Grushin operator We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei.
Paper Structure (4 sections, 5 theorems, 88 equations)

This paper contains 4 sections, 5 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. A functional setting for the Grushin operator
  3. Eigenvalues of the Grushin operator
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $\Omega$ an open bounded subset of $\mathbb{R}^N$, $N\ge 3.$ Let $\lambda\in\mathbb{R}$ and $k \in \mathbb{N}$ be the smallest integer such that $\lambda_{k-1} \leq \lambda < \lambda_k$. Let $\lambda^* = \lambda_k$ and let $m\in\mathbb{N}$ be its multiplicity. Assume that where $S_\gamma$ is the best critical Sobolev constant defined in (def:S) and $\vert \Omega \vert$ denotes the volume of $

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 3.1
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof