Monotonicity and nonexistence results for some fractional elliptic problems in the half space

Abstract

We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space ${\mathbb{R}}_{+}^N:=\{x\in {\mathbb{R}}^N:x_1>0\}$ with the complementary Dirichlet condition $u\equiv 0$ in ${\mathbb{R}}^N\setminus {\mathbb{R}}_{+}^N$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q\ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N\ge 1+2s$. We do not require integrability assumptions on the solutions we study.

24 pages

References in corpus (1)

• On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions

Cited by in corpus (2)

• Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on ${\mathbb{R}}^n$ and ${\mathbb{R}}_{+}^n$
• A Liouville Theorem for a Class of Fractional Systems in ${\mathbb{R}}_{+}^n$