The limit of a nonlocal p-Laplacian obstacle problem with nonhomogeneous term as p → ∞

The aim of this paper is to investigate the asymptotic behavior of the minimizers to the following problems related to the fractional p -Laplacian with nonhomogeneous term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{hₚ(x,u)}} in the presence of an obstacle ψ in a bounded Lipschitz domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:math> {ΩN} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>min</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize="210%" minsize="210%">{</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>×</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> <m:mfrac> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mfrac> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:mi>ψ</m:mi> <m:mo>⁢</m:mo> <m:mtext> on </m:mtext> <m:mo>⁢</m:mo> <m:mover accent="true"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>⁢</m:mo> <m:mtext> on </m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize="210%" minsize="210%">}</m:mo> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> min{\{}1/2∫Ω×Ω{|u(x)-u(y)|ᵖ}{|x-y|% α p}+∫Ω{{{hₚ(x,u)}}}:u∈ Ws,p(Ω),\,u≥ψ% on Ω̄,\,u=g on ∂Ω{\}}. In the case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> <m:mi>p</m:mi> </m:mfrac> </m:mrow> </m:math> {hₚ(x,u)={{h(x,u)}ᵖ}{p}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {h(x,u)≥ 0} , we show the convergence of the solutions to certain limit as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {p→∞} and identify the limit equation. More precisely, we show that the limit problem is closely related to the infinity fractional Laplacian. In the particular case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>s</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {∂ₛh>0} , we study the Hölder regularity of any solution to the limit problem and we extend the existence result to the case when h is not smooth. In addition, we will study the limit of this problem when the nonhomogeneous term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {hₚ(x,u)} is not necessarily positive. To be more precise, we will consider the following two cases: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:math> {hₚ(x,u)=h(x)\,u} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mfrac> <m:msup> <m:mrow> <m:mo lspace="5.8pt" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi mathvariant="normal">Λ</m:mi> </m:msup> <m:mi mathvariant="normal">Λ</m:mi> </m:mfrac> </m:mrow> </m:mrow> </m:math> {hₚ(x,u)=h(x)\,{\,\,|u|Λ}{Λ}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo><</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> {Λ:=Λ(p)<p} .