This work deals with the analysis of eigenvalues, bifurcation and Hölder continuity of solutions to mixed problems like $$ -÷ (|x|-pγ |∇ u|ᵖ⁻²∇ u) = fλ(x,u) , &u > 0\ in Ω ,\\ u = 0 & on Σ_1,\\ |x|-pγ|∇ u|ᵖ⁻²∂ u/∂ ν = 0 & on Σ_2, $$ involving some potentials related with the Caffarelli-Kohn-Nirenberg inequalities, and with different kind of functions $f_λ (x,u)$.
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Colorado et al. (2004) studied this question.