A local bifurcation theorem for degenerate elliptic equations with radial symmetry
The following bifurcation problem is considered \[ -\text{ div }(|\text{ grad }u|^{p-2}\text{ grad }u)=\lambda |u|^{p-2}u+g(\lambda,x,u)\quad\text{ in } B,\quad u=0\quad\text{ on }B, \] where \(p>1\), \(B\subset \mathbb{R}^N\) is a ball with radius \(1\) and centered at the origin, \(g\in C^1\), \(g(\lambda,x,0)\equiv 0\) and \(\partial g/\partial u=o(|u^{p-2}|)\) as \(u\to 0\) and \(\lambda\) is the spectral parameter. Radial solutions are considered only reducing the problem to an ODE. The couple \((\lambda,u)\) is understood as a solution. With \(\{\lambda_{n,p}\}\) being radial eigenvalues of the p-Laplacian on \(B\), main results of the paper can be stated as that the only nontrivial solutions close to the trivial ones \((\lambda_{n,p},0)\) (for \(n,p\) fixed) lie on a continuous curve, thus extending the Crandall-Rabinowitz theorem to degenerated elliptic problems.
degenerate quasilinear elliptic equations
scientific article; zbMATH DE number 1744184
scientific article