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

DOI: http://dx.doi.org/10.12775/TMNA.2004.011

#### Abstract

This work deals with the analysis of

eigenvalues, bifurcation and Hölder continuity of solutions to

mixed problems like

$$

\cases

-\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,

\endcases

$$

involving some potentials related with the

Caffarelli-Kohn-Nirenberg inequalities, and with different

kind of functions $f_\lambda (x,u)$.

eigenvalues, bifurcation and Hölder continuity of solutions to

mixed problems like

$$

\cases

-\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,

\endcases

$$

involving some potentials related with the

Caffarelli-Kohn-Nirenberg inequalities, and with different

kind of functions $f_\lambda (x,u)$.

#### Keywords

Equivalence problems; bifurcation; mixed boundary conditions; Cafarelli-Kohn-Nihrenberg inequalities

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