A semilinear Dirichlet problem involving the fractional Laplacian in R₊ n

We investigate the Dirichelt problem involving the fractional Laplacian in the upper half-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:mfenced close="}" open="{"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">∣</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:mfenced> </m:math> R₊ⁿ=∈ Rⁿ x₁ >0\ : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mtext> </m:mtext> </m:mtd> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="2em" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mtext> </m:mtext> </m:mtd> <m:mtd columnalign="left"> <m:mspace width="2em" /> <m:mspace width="0.3333em" /> <m:mspace width="0.3333em" /> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="2em" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mtext> </m:mtext> </m:mtd> <m:mtd columnalign="left"> <m:mspace width="2em" /> <m:mspace width="0.3333em" /> <m:mspace width="0.3333em" /> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="2em" /> <m:mi>x</m:mi> <m:mo>∉</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> {cases} & {(-{Δ})}ˢu(x)=f(u(x)), x∈ {R}₊ⁿ, \\ & u(x){ >}0, x∈ {R}₊ⁿ, \\ & u(x)=0, x∉ {R}₊ⁿ. {cases}. . We prove the positive solutions are monotonic increasing in the x 1 -direction assuming u ( x ) grows no faster than | x | γ with γ ∈ (0, 2 s ) for | x | large. To start with, we develop a maximum principle on the narrow region. Then we apply a direct method of the moving planes for the fractional Laplacian to derive the monotonicity. As an application of the monotonicity result, we use it to prove nonexistence of bounded positive solutions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> </m:math> R₊ⁿ for f ( u ) = u p , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> </m:math> p∈ (1,n-1+2s/n-1-2s) .