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 ${\mathcal{L}}_{A}u=\mathrm{div}\left(A\left(x\right)\nabla u\right)$ and ${\mathcal{L}}_{a,p}u=\mathrm{div}\left(a\left(x\right){\left|\nabla u\right|}^{p-2}\nabla u\right)$ are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset Ω$\Omega$ of Rⁿ${\mathbb{R}}^n$, 1<p<n$1<p<n$, p^∗=np/(n−p)$p^{\ast }=np/(n-p)$ is the critical Sobolev exponent and λ$\lambda$ is a real parameter. Both problems have been quite studied when the ellipticity of L_A${\mathcal{L}}_A$ and L_a,p${\mathcal{L}}_{a,p}$ concentrate in the interior of Ω$\Omega$. We here focus on the borderline case, namely we assume that the determinant of A(x)$A\left(x\right)$ has a global minimum point 2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀$x_0$ on the boundary of Ω$\Omega$ such that A(x)−A(x₀)$A\left(x\right)-A\left(x_0\right)$ is locally comparable to |x−x₀|^γI_n${|x-x_0|}^{\gamma }{I}_n$ in the bilinear forms sense, where I_n$I_n$ denotes the identity matrix of order n$n$. Similarly, we assume that a352b04a2dac2fcf970e359433c42" title="Click to view the MathML source">a(x)$a\left(x\right)$ has a global minimum point 2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀$x_0$ on the boundary of Ω$\Omega$ such that a(x)−a(x₀)$a\left(x\right)-a\left(x_0\right)$ is locally comparable to |x−x₀|^σ${|x-x_0|}^{\sigma }$. We provide a linking between the exponents γ$\gamma$ and σ$\sigma$ and the order of singularity of the boundary of Ω$\Omega$ at 2a30fd6ed3dcd804baf016f1fb" title="Click to view the MathML source">x₀$x_0$ so that these problems admit at least one positive solution for any 32a7d6a0fa1b3" title="Click to view the MathML source">λ∈(0,λ₁(−L_A))$\lambda \in (0,{\lambda }_1(-{\mathcal{L}}_A))$ and λ∈(0,λ₁(−L_a,p))$\lambda \in (0,{\lambda }_1(-{\mathcal{L}}_{a,p}))$, respectively, where 2a3b638c8b16634b" title="Click to view the MathML source">λ₁${\lambda }_1$ denotes the first Dirichlet eigenvalue of the corresponding operator.