Bifurcation points of a degenerate elliptic boundary-value problem
The authors study the nonlinear degenerate elliptic boundary value problem \[ - \nabla [A(x) \nabla u(x)]= \lambda f( u(x) ),\quad x \in \Omega \] \[ u(x)=0,\qquad x\in \partial \Omega \] \(\Omega\) being a bounded open subset of \(\mathbb R^N\), \(N \geq 3,\) having Lipschitz boundary. The ellipticity is degenerate in the sense that \(0\in\Omega \) and \(A(x)>0 \) for all \(x\) not equal to zero, but \(\lim_{x \to 0}\frac{A(x)}{x^2}=1\). In the note it is proved that there is a vertical bifurcation in all points \(\lambda\in [N^2/4,\infty)\) that also occurs at all the eigenvalues of the linearized problem that are below \(N^2/4\). We point out that the approach is based on a recent study by the authors concerning problems that are differentiable in the sense of Hadamard, but not in the sense of Fréchet.
scientific article; zbMATH DE number 5173046
scientific article