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- W2962834853 abstract "This note concerns the existence of positive solutions for the boundary problems {−LAu=u2∗−1+λuinΩu=0on∂Ω,{−La,pu=up∗−1+λup−1inΩu=0on∂Ω,where LAu=div(A(x)∇u) and La,pu=div(a(x)|∇u|p−2∇u) are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset ΩΩ of RnRn, 1<p<n1<p<n, p∗=np/(n−p)p∗=np/(n−p) is the critical Sobolev exponent and λλ is a real parameter. Both problems have been quite studied when the ellipticity of LALA and La,pLa,p concentrate in the interior of ΩΩ. We here focus on the borderline case, namely we assume that the determinant of A(x)A(x) has a global minimum point x0x0 on the boundary of ΩΩ such that A(x)−A(x0)A(x)−A(x0) is locally comparable to |x−x0|γIn|x−x0|γIn in the bilinear forms sense, where InIn denotes the identity matrix of order nn. Similarly, we assume that a(x)a(x) has a global minimum point x0x0 on the boundary of ΩΩ such that a(x)−a(x0)a(x)−a(x0) is locally comparable to |x−x0|σ|x−x0|σ. We provide a linking between the exponents γγ and σσ and the order of singularity of the boundary of ΩΩ at x0x0 so that these problems admit at least one positive solution for any λ∈(0,λ1(−LA))λ∈(0,λ1(−LA)) and λ∈(0,λ1(−La,p))λ∈(0,λ1(−La,p)), respectively, where λ1λ1 denotes the first Dirichlet eigenvalue of the corresponding operator." @default.
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- W2962834853 date "2016-12-01" @default.
- W2962834853 modified "2023-10-12" @default.
- W2962834853 title "A note on borderline Brezis–Nirenberg type problems" @default.
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- W2962834853 doi "https://doi.org/10.1016/j.na.2016.09.003" @default.
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