The Brezis-Nirenberg result for the fractional Laplacian

2014/09/22 by Raffaella Servadei, Enrico Valdinoci · 28 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.1090/s0002-9947-2014-05884-4

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"> \ \begin arrayll (-Δ )^s u-λ u=|u|^{2^*-2}u amp; \mbox in Ω ,
u=0 amp; \mbox in \mathbb Rⁿ∖ Ω , \end array .</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">(-Δ )^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">s∈ (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">Ω</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>

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• On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''
• Semilinear equations involving the fractional Laplacian on domains
• Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian
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• A priori bounds and existence of solutions for some nonlocal elliptic problems
• Existence results for fractional p-Laplacian problems via Morse theory
• Fractional Elliptic problem with Finite many critical Hardy--Sobolev Exponents
• Existence, Uniqueness of Positive Solution to a Fractional Laplacians with Singular Nonlinearity
• Non variational type critical growth nonlocal system
• Fractional p-Laplacian Kirchhoff-type problem involving a singular term via Nehari manifold
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• Minimal energy solutions to the fractional Lane-Emden system, I: Existence and singularity formation
• Normalized solutions for fractional Choquard equation with critical growth on bounded domain
• Bifurcation and multiplicity results for critical fractional p-Laplacian problems
• Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method
• The Nehari manifold for fractional p-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions
• Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group
• Unbalanced (p,2)-fractional problems with critical growth
• On doubly nonlocal p-fractional coupled elliptic system
• Fractional Poincar 'e Inequality for Unbounded Domains with Finite Ball\n Condition: Counter Example
• Asymptotic behavior of solutions for nonlinear elliptic problems with\n the fractional Laplacian
• Stationary Kirchhoff problems involving a fractional elliptic operator\n and a critical nonlinearity
• Certain fractional Laplacian equations that do not have smooth solutions
• Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities
• Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters. [europepmc]

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