Regularity of radial extremal solutions for some non local semilinear equations

Abstract

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-Δ)^s u&=λf(u) \qquad {\mbox{ in $B_1\subset {\mathbb{R}}^n$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge 2$, $s\in (0,1)$, $\lambda \ge 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta {)}^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$.

References in corpus (2)

• Regularity of minimizers of semilinear elliptic problems up to dimension four
• Positive solutions of nonlinear problems involving the square root of the Laplacian

Cited by in corpus (1)

• Singular solutions of fractional elliptic equations with absorption