A note on borderline Brezis-Nirenberg type problems

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• 作者：Julian Haddad ; ^{julianhaddad@ufmg.br" class="auth_mail" title="E-mail the corresponding author} ; Marcos Montenegro ^{montene@mat.ufmg.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J25 ; secondary ; 35J20 ; 35J60 ; 35J62 ; 35J66
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：147
• 期：Complete
• 页码：169-175
• 全文大小：595 K

文摘

This note concerns the existence of positive solutions for the boundary problems where an id="mmlsi3" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si3.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=60f79b766009dec8ed56de558a3a276c">ass="imgLazyJSB inlineImage" height="13" width="119" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302048-si3.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">athvariant="script">LAu=athvariant="normal">div(A(x)&nabla;u)ath>an>an>an> and an id="mmlsi4" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=a01489f9bde5915a69a9d3527276db5a">ass="imgLazyJSB inlineImage" height="16" width="172" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302048-si4.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">athvariant="script">La,pu=athvariant="normal">div(a(x)|&nabla;u|p−2&nabla;u)ath>an>an>an> are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=6b70ff00808b0b39cfcc1a542e5a5de5" title="Click to view the MathML source">Ωan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Ωath>an>an>an> of an id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si6.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=ea8e5ea68f1943cfd0ad41e35199dc08" title="Click to view the MathML source">Rⁿan>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">athvariant="double-struck">Rnath>an>an>an>, an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si7.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=835cf257c29f22c8eb0e8a2ebb4e7856" title="Click to view the MathML source">1<p<nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">1<p<nath>an>an>an>, an id="mmlsi8" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si8.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=50a45b8ac6f9f344f47a75d68e2ccdf6" title="Click to view the MathML source">p^&lowast;=np/(n−p)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">p&lowast;=np/(n−p)ath>an>an>an> is the critical Sobolev exponent and an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si9.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=97f06d79254e9175d3714831baa64e4d" title="Click to view the MathML source">λan>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">λath>an>an>an> is a real parameter. Both problems have been quite studied when the ellipticity of an id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si10.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=f8fbf0fc6b3d1a10cb14c38abd21c6db" title="Click to view the MathML source">L_Aan>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">athvariant="script">LAath>an>an>an> and an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si11.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=618ffeebaf8dd854310ca15e8febf935" title="Click to view the MathML source">L_a,pan>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">athvariant="script">La,path>an>an>an> concentrate in the interior of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=6b70ff00808b0b39cfcc1a542e5a5de5" title="Click to view the MathML source">Ωan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Ωath>an>an>an>. We here focus on the borderline case, namely we assume that the determinant of an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si13.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=bfe6f3ad7fd992710d1f78573b2c9e67" title="Click to view the MathML source">A(x)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">A(x)ath>an>an>an> has a global minimum point an id="mmlsi14" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si14.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=da842c2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">x0ath>an>an>an> on the boundary of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=6b70ff00808b0b39cfcc1a542e5a5de5" title="Click to view the MathML source">Ωan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Ωath>an>an>an> such that an id="mmlsi16" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si16.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=5ce9cf233f14d7ddbeaa782011c84172" title="Click to view the MathML source">A(x)−A(x₀)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si16.gif" overflow="scroll">A(x)−A(x0)ath>an>an>an> is locally comparable to an id="mmlsi17" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si17.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=cc951f76a7fed64f7e031cb167d56d92" title="Click to view the MathML source">|x−x₀|^γI_nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si17.gif" overflow="scroll">|x−x0|γInath>an>an>an> in the bilinear forms sense, where an id="mmlsi18" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si18.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=9638d2281e17afda2202e1548c478ebc" title="Click to view the MathML source">I_nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si18.gif" overflow="scroll">Inath>an>an>an> denotes the identity matrix of order an id="mmlsi19" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si19.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=430939c68ca4438e8c34622433a9a6b9" title="Click to view the MathML source">nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si19.gif" overflow="scroll">nath>an>an>an>. Similarly, we assume that an id="mmlsi20" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si20.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=843a352b04a2dac2fcf970e359433c42" title="Click to view the MathML source">a(x)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si20.gif" overflow="scroll">a(x)ath>an>an>an> has a global minimum point an id="mmlsi14" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si14.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=da842c2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">x0ath>an>an>an> on the boundary of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=6b70ff00808b0b39cfcc1a542e5a5de5" title="Click to view the MathML source">Ωan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Ωath>an>an>an> such that an id="mmlsi23" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si23.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=a4d491c5039c43a7bc113459f8774ed5" title="Click to view the MathML source">a(x)−a(x₀)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si23.gif" overflow="scroll">a(x)−a(x0)ath>an>an>an> is locally comparable to an id="mmlsi24" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si24.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=ab50b33f585c98dee77e1b48970478ff" title="Click to view the MathML source">|x−x₀|^σan>an class="mathContainer hidden">an class="mathCode">ath altimg="si24.gif" overflow="scroll">|x−x0|σath>an>an>an>. We provide a linking between the exponents an id="mmlsi25" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si25.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=d78093d818a191ba499db8f35f20629f" title="Click to view the MathML source">γan>an class="mathContainer hidden">an class="mathCode">ath altimg="si25.gif" overflow="scroll">γath>an>an>an> and an id="mmlsi26" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si26.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=4c1168543ce502aefd43cb71f16128e4" title="Click to view the MathML source">σan>an class="mathContainer hidden">an class="mathCode">ath altimg="si26.gif" overflow="scroll">σath>an>an>an> and the order of singularity of the boundary of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=6b70ff00808b0b39cfcc1a542e5a5de5" title="Click to view the MathML source">Ωan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Ωath>an>an>an> at an id="mmlsi14" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si14.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=da842c2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">x0ath>an>an>an> so that these problems admit at least one positive solution for any an id="mmlsi29" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si29.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=f88b50253f9b41b13ae32a7d6a0fa1b3" title="Click to view the MathML source">λ∈(0,λ₁(−L_A))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si29.gif" overflow="scroll">λ∈(0,λ1(−athvariant="script">LA))ath>an>an>an> and an id="mmlsi30" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si30.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=ed002cdd05ce64fbe5b33d0a0d84c1d0" title="Click to view the MathML source">λ∈(0,λ₁(−L_a,p))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si30.gif" overflow="scroll">λ∈(0,λ1(−athvariant="script">La,p))ath>an>an>an>, respectively, where an id="mmlsi31" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302048&_mathId=si31.gif&_user=111111111&_pii=S0362546X16302048&_rdoc=1&_issn=0362546X&md5=9ee6245801f216e42a3b638c8b16634b" title="Click to view the MathML source">λ₁an>an class="mathContainer hidden">an class="mathCode">ath altimg="si31.gif" overflow="scroll">λ1ath>an>an>an> denotes the first Dirichlet eigenvalue of the corresponding operator.