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W2049434583 abstract "The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis negative normal upper Delta right-parenthesis Superscript s Baseline u minus lamda u equals StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript asterisk Superscript minus 2 Baseline u 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon in double-struck upper R Superscript n Baseline minus normal upper Omega comma EndLayout> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=left left rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:mo stretchy=false>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext> in </mml:mtext> </mml:mstyle> </mml:mrow> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext> in </mml:mtext> </mml:mstyle> </mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo class=MJX-variant>∖</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>left { begin {array}{ll} (-Delta )^s u-lambda u=|u|^{2^*-2}u & {mbox { in }} Omega , u=0 & {mbox { in }} mathbb {R}^nsetminus Omega ,, end {array} right .</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis negative normal upper Delta right-parenthesis Superscript s> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>(-Delta )^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the fractional Laplace operator, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s element-of left-parenthesis 0 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sin (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an open bounded set of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than 2 s> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>n>2s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with Lipschitz boundary, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a real parameter and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 Superscript asterisk Baseline equals 2 n slash left-parenthesis n minus 2 s right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>2^*=2n/(n-2s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout Enlarged left-brace 1st Row 1st Column script upper L Subscript upper K Baseline u plus lamda u plus StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript asterisk Superscript minus 2 Baseline u plus f left-parenthesis x comma u right-parenthesis equals 0 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon in double-struck upper R Superscript n Baseline minus normal upper Omega comma EndLayout> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=left left rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>in </mml:mtext> </mml:mstyle> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>in </mml:mtext> </mml:mstyle> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo class=MJX-variant>∖</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>left { begin {array}{ll} mathcal L_K u+lambda u+|u|^{2^*-2}u+f(x, u)=0 & mbox {in } Omega , u=0 & mbox {in } mathbb {R}^nsetminus Omega ,, end {array}right .</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L Subscript upper K> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal L_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a general non-local integrodifferential operator of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a lower order perturbation of the critical power <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript asterisk Superscript minus 2 Baseline u> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>|u|^{2^*-2}u</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda Subscript 1 comma s> <mml:semantics> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>lambda _{1,s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the first eigenvalue of the non-local operator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis negative normal upper Delta right-parenthesis Superscript s> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>(-Delta )^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with homogeneous Dirichlet boundary datum, then for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda element-of left-parenthesis 0 comma lamda Subscript 1 comma s Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda in (0, lambda _{1,s})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a non-trivial solution of the above model equation, provided <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-slanted-equals 4 s> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>ngeqslant 4s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators." @default.
W2049434583 created "2016-06-24" @default.
W2049434583 creator A5078447699 @default.
W2049434583 creator A5086859934 @default.
W2049434583 date "2014-09-22" @default.
W2049434583 modified "2023-10-16" @default.
W2049434583 title "The Brezis-Nirenberg result for the fractional Laplacian" @default.
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