Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet-Neumann boundary conditions related to Caffarelli-Kohn-Nirenberg inequalities
This paper is devoted to the analysis of eigenvalues, bifurcation and Hölder continuity of solutions to mixed problems like \[ \begin{aligned} -\text{div}(|x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)= f_1(x,u),\quad u> 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\Sigma_1,\\ |x|^{-p\gamma}|\nabla u|^{p-2}{\partial u\over\partial\nu}= 0\quad &\text{on }\Sigma_2,\end{aligned} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\) with \(0\in\Omega\), \(\Sigma_1\) and \(\Sigma_2\) are smooth \((N-1)\)-dimensional submanifolds of \(\partial\Omega\) such that \(\Sigma_1\cap\Sigma_2= \emptyset\), \(\overline\Sigma_1\cup \overline\Sigma_2= \partial\Omega\), \(\overline\Sigma_1\cap\overline\Sigma_2= \Gamma\) is a smooth \((N-2)\)-dimensional submanifold; \(-\infty< \gamma<{N-p\over p}\). The main feature of the paper is to consider mixed boundary problems with weights associated to Caffarelli-Kohn-Nirenberg inequalities.
scientific article; zbMATH DE number 2157975