Strong Comparison Principle for the Fractional p-Laplacian and Applications to Starshaped Rings

Abstract

Abstract In the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze { ( - Δ ) p s ⁢ v + q ⁢ ( x ) ⁢ | v | p - 2 ⁢ v ≥ 0 in D , ( - Δ ) p s ⁢ w + q ⁢ ( x ) ⁢ | w | p - 2 ⁢ w ≤ 0 in D , v ≥ w in ℝ N , \quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , p > 1 {p>1} , D ⊂ ℝ N {D\subset\mathbb{R}^{N}} is an open set, and q ∈ L ∞ ⁢ ( ℝ N ) {q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either v ≡ w {v\equiv w} in ℝ N {\mathbb{R}^{N}} or v > w {v>w} in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.