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

Abstract

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta {)}_p^{s}u=\lambda f(u),u>0\text{in}\Omega ;u=0\text{in}{\mathbb{R}}^N\setminus \Omega$, where $p>1$, $s\in (0,1)$, $\lambda >0$ and $\Omega$ 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 ${\lambda }^{\ast }\in (0,\infty )$ such that a bounded minimal solution $u_{\lambda }\in W_0^{s,p}(\Omega )$ exists if $\lambda \in (0,{\lambda }^{\ast })$, and no bounded solution exists if $\lambda >{\lambda }^{\ast }$. Moreover, no $W_0^{s,p}(\Omega )$ solution exists for $\lambda >{\lambda }^{\ast }$ 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 $(-\Delta {)}_p^{s}u=g$ with $g\in W_0^{s,p}(\Omega {)}^{\ast }\cap L^q(\Omega )$ for $q\ge 1$, especially for $q\le \frac{N}{sp}$. We believe that these general results have their own interests. Finally, using the stability of minimal solutions $u_{\lambda }$, under the polynomial growth or convexity assumption on $f$, we show that the extremal function $u_{\ast }={\lim }_{\lambda \to {\lambda }^{\ast }}{u}_{\lambda }\in W_0^{s,p}(\Omega )$ in all dimensions, and $u^{\ast }\in L^{\infty }(\Omega )$ in some low dimensional cases.

References in corpus (6)

• The second eigenvalue of the fractional $p-$Laplacian
• Nonlocal equations with measure data
• A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian
• Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities
• Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian
• A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations