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: $(-\hat{I}”{)}_p^{s}u=\hat{I}»f(u),u>0\text{in}\hat{I}\text{c◯};u=0\text{in}{\mathbb{R}}^N\setminus \hat{I}\text{c◯}$, where $p>1$, $s\in (0,1)$, $\hat{I}»>0$ and $\hat{I}\text{c◯}$ 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 $\hat{I}{»}^{\ast }\in (0,\infty )$ such that a bounded minimal solution $u_{\hat{I}}»\in W_0^{s,p}(\hat{I}\text{c◯})$ exists if $\hat{I}»\in (0,\hat{I}{»}^{\ast })$, and no bounded solution exists if $\hat{I}»>\hat{I}{»}^{\ast }$. Moreover, no $W_0^{s,p}(\hat{I}\text{c◯})$ solution exists for $\hat{I}»>\hat{I}{»}^{\ast }$ 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 $(-\hat{I}”{)}_p^{s}u=g$ with $g\in W_0^{s,p}(\hat{I}\text{c◯}{)}^{\ast }\cap L^q(\hat{I}\text{c◯})$ 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 $u_{\hat{I}}»$, under the polynomial growth or convexity assumption on $f$, we show that the extremal function $u_{\ast }={\lim }_{\hat{I}»\to \hat{I}{»}^{\ast }}{u}_{\hat{I}}»\in W_0^{s,p}(\hat{I}\text{c◯})$ in all dimensions, and $u^{\ast }\in L^{\infty }(\hat{I}\text{c◯})$ in some low dimensional cases.