A note on borderline Brezis–Nirenberg type problems

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 Ω of Rn, 1<p<n, p∗=np/(n−p) is the critical Sobolev exponent and λ is a real parameter. Both problems have been quite studied when the ellipticity of LA and La,p concentrate in the interior of Ω. We here focus on the borderline case, namely we assume that the determinant of A(x) has a global minimum point x0 on the boundary of Ω such that A(x)−A(x0) is locally comparable to |x−x0|γIn in the bilinear forms sense, where In denotes the identity matrix of order n. Similarly, we assume that a(x) has a global minimum point x0 on the boundary of Ω such that a(x)−a(x0) is locally comparable to |x−x0|σ. We provide a linking between the exponents γ and σ and the order of singularity of the boundary of Ω at x0 so that these problems admit at least one positive solution for any λ∈(0,λ1(−LA)) and λ∈(0,λ1(−La,p)), respectively, where λ1 denotes the first Dirichlet eigenvalue of the corresponding operator.