The number of positive solutions to the Brezis-Nirenberg problem
Daomin Cao, Peng Luo, Shuangjie Peng
Abstract
In this paper we are concerned with the well-known Brezis-Nirenberg problem <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 minus normal upper Delta u equals u Superscript StartFraction upper N plus 2 Over upper N minus 2 EndFraction Baseline plus epsilon u comma 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u greater-than 0 comma 2nd Column a m p semicolon in normal upper Omega comma 3rd Row 1st Column u equals 0 comma 2nd Column a m p semicolon on partial-differential normal upper Omega period EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> </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:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <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:mo>,</mml:mo> </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:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <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:mo>,</mml:mo> </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:mtext>on</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <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{equation*} \begin {cases} -\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, &{\text {in}~\Omega },\\ u>0, &{\text {in}~\Omega },\\ u=0, &{\text {on}~\partial \Omega }. \end{cases} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> The existence of multi-peak solutions to the above problem for small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was obtained (see Monica Musso and Angela Pistoia [Indiana Univ. Math. J. 51 (2002), pp. 541–579]). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green’s function of the domain <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> and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi>ε</mml:mi> <mml:annotation encoding="application/x-tex">\varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which depends also on Green’s function.