On Dirichlet problem for fractional <i>p</i> -Laplacian with singular non-linearity

Abstract

Abstract In this article, we study the following fractional p -Laplacian equation with critical growth and singular non-linearity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="22.5pt">,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>∖</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u&gt;0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} with smooth boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> {\partial\Omega} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> {n&gt;sp} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {s\in(0,1)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>λ</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {\lambda&gt;0} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <