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 class="mathmlsrc">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">class="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">class="mathContainer hidden">class="mathCode">${\mathcal{L}}_{A}u=\mathrm{div}\left(A\left(x\right)\nabla u\right)$ and class="mathmlsrc">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">class="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">class="mathContainer hidden">class="mathCode">${\mathcal{L}}_{a,p}u=\mathrm{div}\left(a\left(x\right){\left|\nabla u\right|}^{p-2}\nabla u\right)$ are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset class="mathmlsrc">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">Ωclass="mathContainer hidden">class="mathCode">$\Omega$ of class="mathmlsrc">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">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$, class="mathmlsrc">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<nclass="mathContainer hidden">class="mathCode">$1<p<n$, class="mathmlsrc">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^∗=np/(n−p)class="mathContainer hidden">class="mathCode">$p^{\ast }=np/(n-p)$ is the critical Sobolev exponent and class="mathmlsrc">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">λclass="mathContainer hidden">class="mathCode">$\lambda$ is a real parameter. Both problems have been quite studied when the ellipticity of class="mathmlsrc">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_Aclass="mathContainer hidden">class="mathCode">${\mathcal{L}}_A$ and class="mathmlsrc">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,pclass="mathContainer hidden">class="mathCode">${\mathcal{L}}_{a,p}$ concentrate in the interior of class="mathmlsrc">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">Ωclass="mathContainer hidden">class="mathCode">$\Omega$. We here focus on the borderline case, namely we assume that the determinant of class="mathmlsrc">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)class="mathContainer hidden">class="mathCode">$A\left(x\right)$ has a global minimum point class="mathmlsrc">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">x₀class="mathContainer hidden">class="mathCode">$x_0$ on the boundary of class="mathmlsrc">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">Ωclass="mathContainer hidden">class="mathCode">$\Omega$ such that class="mathmlsrc">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)class="mathContainer hidden">class="mathCode">$A\left(x\right)-A\left(x_0\right)$ is locally comparable to class="mathmlsrc">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_nclass="mathContainer hidden">class="mathCode">${|x-x_0|}^{\gamma }{I}_n$ in the bilinear forms sense, where class="mathmlsrc">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_nclass="mathContainer hidden">class="mathCode">$I_n$ denotes the identity matrix of order class="mathmlsrc">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">nclass="mathContainer hidden">class="mathCode">$n$. Similarly, we assume that class="mathmlsrc">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)class="mathContainer hidden">class="mathCode">$a\left(x\right)$ has a global minimum point class="mathmlsrc">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">x₀class="mathContainer hidden">class="mathCode">$x_0$ on the boundary of class="mathmlsrc">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">Ωclass="mathContainer hidden">class="mathCode">$\Omega$ such that class="mathmlsrc">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)class="mathContainer hidden">class="mathCode">$a\left(x\right)-a\left(x_0\right)$ is locally comparable to class="mathmlsrc">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|^σclass="mathContainer hidden">class="mathCode">${|x-x_0|}^{\sigma }$. We provide a linking between the exponents class="mathmlsrc">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">γclass="mathContainer hidden">class="mathCode">$\gamma$ and class="mathmlsrc">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">σclass="mathContainer hidden">class="mathCode">$\sigma$ and the order of singularity of the boundary of class="mathmlsrc">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">Ωclass="mathContainer hidden">class="mathCode">$\Omega$ at class="mathmlsrc">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">x₀class="mathContainer hidden">class="mathCode">$x_0$ so that these problems admit at least one positive solution for any class="mathmlsrc">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))class="mathContainer hidden">class="mathCode">$\lambda \in (0,{\lambda }_1(-{\mathcal{L}}_A))$ and class="mathmlsrc">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))class="mathContainer hidden">class="mathCode">$\lambda \in (0,{\lambda }_1(-{\mathcal{L}}_{a,p}))$, respectively, where class="mathmlsrc">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">λ₁class="mathContainer hidden">class="mathCode">${\lambda }_1$ denotes the first Dirichlet eigenvalue of the corresponding operator.