[Paper Review] High-order Kirchhoff problems in bounded and unbounded domains

Mohamed Karim Hamdani, Abdellaziz Harrabi|arXiv (Cornell University)|Jul 29, 2018

Advanced Mathematical Modeling in Engineering25 references4 citations

TL;DR

This paper establishes the existence of infinitely many solutions for high-order Kirchhoff problems involving the m-polyharmonic operator in bounded and unbounded domains, including the zero mass case on ℝᴺ. By introducing a novel quantity λₘ analogous to the first eigenvalue and employing a Schauder basis in W₀ʳᵐ(Ω), the authors verify the symmetric mountain pass geometry without requiring control of f near zero, even under weak growth conditions on f at infinity, extending variational methods to degenerate and unbounded settings.

ABSTRACT

Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray} \label{ea} \begin{cases} M\left(\int_Ø|D_r u|^{m} +a|u|^m ight)[Δ^r_m u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad Ω, \\ u=\left(\frac{\partial}{\partial ν} ight)^k u=0, \quad &\mbox{on}\quad \partialΩ, \quad k=1, 2,..... , r-1, \end{cases} \end{eqnarray} where $r \in \N^*$, $m >1$, $N\geq rm+1$, $a\geq 0$, $K\in L^{\infty}(Ø)$ is a positive weight function, $M \in C([0,+\infty))$ and $f\in C(\mathbb{R})$ which will be specified later. We will study problem \eqref{ea} in the following different type of domains: \begin{enumerate} \item $a=0$ and $K\in L^{\infty}(Ø)$ is a positive weight function if $Ω$ is a smooth bounded domain of $\R^N$. \item $a>0$ and $K\in L^{\infty}(Ø)\cap L^{p}(Ø)$, $p \geq 1$ if $Ω$ is an unbounded smooth domain. \item $Ø=\R^N$ and $a=0$ (which called the $mγ$-zero mass case). \end{enumerate} We prove the existence of infinitely many solutions of \eqref{ea} for some odd functions $f$ in $u$ satisfying subcritical growth conditions at infinity which are weaker than the analogue of the Ambrosetti-Rabinowitz condition and the standard subcritical polynomial growth. The new aspect consists in employing the Schauder basis of $W_0^{r,m}(Ø)$ to verify the geometry of the symmetric mountain pass theorem without any control on $f$ near $0$ if $Ω$ is a bounded domain and under a suitable condition at $0$ if $Ω$ is a unbounded domain allowing only to derive the variational setting of \eqref{ea}. Moreover, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution.

Motivation & Objective

• To establish the existence of infinitely many solutions for high-order Kirchhoff problems involving the m-polyharmonic operator in bounded and unbounded domains.
• To extend variational methods to degenerate Kirchhoff problems where M(τ) may vanish at τ=0, overcoming the lack of coercivity.
• To analyze the zero mass case (a=0, Ω=ℝᴺ) where standard Sobolev embeddings fail, requiring new functional settings.
• To develop a variational framework for non-degenerate and degenerate m-polyharmonic Kirchhoff problems under weak growth conditions on f.
• To introduce a new quantity λₘ resembling the first eigenvalue of the m-polyharmonic operator to construct mountain pass solutions.

Proposed method

• Define the m-polyharmonic operator Δʳₘ via iterated applications of the m-Laplacian, distinguishing between odd and even r.
• Introduce the space E = W₀ʳᵐ(Ω) with norm ||u||ᵐ = ∫(|Dʳu|ᵐ + a|u|ᵐ), and Dʳu as the r-th order differential operator.
• Define λₘ as the infimum of ||u||ᵐ / ∫|u|ᵐᵞ over u ∈ E extbackslash {0}, analogous to a first eigenvalue, used to control the energy functional.
• Use a Schauder basis of W₀ʳᵐ(Ω) to verify the symmetric mountain pass geometry without requiring f to be controlled near zero.
• Apply the symmetric mountain pass theorem and the (PS) or (C) condition depending on m ≥ 2 or 1 < m < 2, under assumptions (M₁)–(M₃) on M and (H₂), (H₃′) on f.
• Establish mountain pass geometry by proving I(u) ≥ α > 0 on ||u|| = ρ and I(tφ) < 0 for large t, using the new λₘ and weak growth conditions on f.

Experimental results

Research questions

• RQ1 Can the symmetric mountain pass theorem be applied to high-order Kirchhoff problems in unbounded domains without requiring f to be controlled near zero?
• RQ2 How can the variational framework be extended to degenerate Kirchhoff problems where M(τ) may vanish at τ=0?
• RQ3 What is the role of the new quantity λₘ in constructing mountain pass solutions for m-polyharmonic Kirchhoff problems?
• RQ4 How can the Schauder basis of W₀ʳᵐ(Ω) be used to verify the symmetric mountain pass geometry without control of f near zero?
• RQ5 Can the zero mass case (a=0, Ω=ℝᴺ) be treated variationaly using a new functional setting and eigenvalue-like quantity λₘ?

Key findings

• The paper proves the existence of infinitely many solutions for the m-polyharmonic Kirchhoff problem in bounded and unbounded domains under weak growth conditions on f.
• The existence result holds without requiring f to be controlled near zero when Ω is bounded, thanks to the use of a Schauder basis in W₀ʳᵐ(Ω).
• For unbounded domains, a suitable condition on f at zero is required, but the method avoids strong assumptions like the Ambrosetti-Rabinowitz condition.
• The introduction of λₘ, a quantity analogous to the first eigenvalue of the m-polyharmonic operator, enables the construction of mountain pass solutions.
• The mountain pass geometry is verified via a new energy estimate: I(u) ≥ α > 0 on ||u|| = ρ and I(tφ) < 0 for large t, using λₘ and weak growth of f.
• The Palais-Smale or Cerami condition is satisfied under the assumptions (M₁)–(M₃), ensuring the existence of a critical point for the energy functional.