where and 3527276db5a"> are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset Ω of 35199dc08" title="Click to view the MathML source">Rn, 35cf257c29f22c8eb0e8a2ebb4e7856" title="Click to view the MathML source">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 35" title="Click to view the MathML source">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 352b04a2dac2fcf970e359433c42" title="Click to view the MathML source">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 35f20629f" title="Click to view the MathML source">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.