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

Abstract 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> {{h_{p}(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> {\Omega\subset\mathbb{R}^{N}} , <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