Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet-Neumann boundary conditions related to Caffarelli-Kohn-Nirenberg inequalities

Abstract

This work deals with the analysis of eigenvalues, bifurcation and Hölder continuity of solutions to mixed problems like $$ \begin{cases} -{\rm div} (|x|^{-p\gamma} |\nabla u|^{p-2}\nabla u) = f_{\lambda}(x,u) , &u > 0\ \text{ in }\Omega ,\\ u = 0 &\text{ on }\Sigma_1,\\ |x|^{-p\gamma}|\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu} = 0 &\text{ on } \Sigma_2, \end{cases} $$ involving some potentials related with the Caffarelli-Kohn-Nirenberg inequalities, and with different kind of functions $f_\lambda (x,u)$.