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- W2962788670 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>></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>0quadtext{in }Omega,% qquad u=0quadtext{in }mathbb{R}^{n}setminusOmega, 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> {partialOmega} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>n</m:mi> <m:mo>></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>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> {sin(0,1)} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {lambda>0} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<qleq 1} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>≤</m:mo> <m:msubsup> <m:mi>p</m:mi> <m:mi>s</m:mi> <m:mo>*</m:mo> </m:msubsup> </m:mrow> </m:math> {1<p<alpha+1leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ." @default.
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- W2962788670 date "2016-12-02" @default.
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- W2962788670 title "On Dirichlet problem for fractional <i>p</i>-Laplacian with singular non-linearity" @default.
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