Stable solution and extremal solution for fractional $p$-Laplacian

TL;DR

This work extends the theory of elliptic dichotomy and extremal solutions to the nonlocal, nonlinear regime of the fractional $p$-Laplacian. By introducing a notion of stable solutions and proving a Kato-type inequality, the authors establish a dichotomy at a critical parameter $\lambda^*$: bounded minimal solutions exist for $0<λ<λ^*$ and no bounded solutions exist for $λ>λ^*$. They develop robust $L^r$- and regularity estimates for the fractional operator, prove extremal limits $u^*\in W_0^{s,p}(Ω)$ (and potentially $L^∞(Ω)$ in low dimensions) under power-like or convex nonlinearities, and extend classical local results to the nonlocal setting, providing a comprehensive framework for existence, stability, and regularity of extremal solutions in fractional nonlinear problems.

Abstract

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-Δ)_p^s u= λf(u),\; u> 0 ~\text{in}~Ω;\; u=0\;\text{in}~ \mathbb{R}^N\setminusΩ$, where $p>1$, $s\in (0,1)$, $λ>0$ and $Ω$ is a bounded domain with $C^{1, 1}$ boundary. We first propose a notion of stable solution, then we prove that when $f$ is of class $C^1$, nondecreasing and satisfying $f(0)>0$ and $\underset{t\to \infty}{\lim}\frac{f(t)}{t^{p-1}}=\infty$, there exists an extremal parameter $λ^*\in (0, \infty)$ such that a bounded minimal solution $u_λ\in W_0^{s,p}(Ω)$ exists if $λ\in (0, λ^*)$, and no bounded solution exists if $λ>λ^*$. Moreover, no $W_0^{s,p}(Ω)$ solution exists for $λ> λ^*$ if in addition $f(t)^{\frac{1}{p-1}}$ is convex. To handle our problems, we show a Kato-type inequality for fractional $p$-Laplacian. We show also $L^r$ estimates for the equation $(-Δ)_p^su=g$ with $g\in W_0^{s, p}(Ω)^*\cap L^q(Ω)$ for $q \geq 1$, especially for $q \le \frac{N}{sp}$. We believe that these general results have their own interests. Finally, using the stability of minimal solutions $u_λ$, under the polynomial growth or convexity assumption on $f$, we show that the extremal function $u_* =\lim_{λ\toλ^*}u_λ\in W_0^{s,p}(Ω)$ in all dimensions, and $u^*\in L^{\infty}(Ω)$ in some low dimensional cases.

Theorems & Definitions (38)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10