Bifurcation theory for semilinear elliptic boundary value problems
The paper is devoted to static bifurcation theory for a class of semilinear elliptic degenerate boundary value problem: \[ \begin{cases}-\text{div}(A(x)\nabla u)+c(x)u=f(\lambda ,x,u) & \text{in }D \\ Bu:=a\frac{\partial u}{\partial v}+bu=\varphi & \text{on }\partial D. \end{cases} \] Here \(D\) is a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary \(\partial D,\) \(u\in C^{2+\theta }(D)\) for some \(\theta \in (0,1)\), \(\lambda \) is a real parameter, \(c(x)>0\) in \(D,\) and \(A(x)\) is a symmetric matrix with real smooth coefficients for which there exists a constant \(a_{0}>0\) such that \((A(x)\xi ,\xi)\geq a_{0}|\xi |^{2},\) \(x\in \overline{D},\) \(\xi \in \mathbb{R}^{N}.\) The word degenerate means that the smooth boundary functions \(a\) and \(b\) satisfy: \(a\geq 0\) on \(\partial D\), \(b\geq 0\) on \(\partial D\), and \(b>0\) on \(M=\{\sigma \in \partial D\): \(a(\sigma)=0\}.\) At first, it is shown that the first eigenvalue of the operator \({\mathcal A}u=-\text{div}(A(x)\nabla u)+c(x)u,\) \(u\in {\mathcal D(A)}=\{u\in W^{2,2}(D):Bu=0\}\) is simple with positive eigenfunction. The given spectral analysis of the operator \({\mathcal A}\) is the background for the proof of the local static bifurcation theorem for (1), where \(f(\lambda ,x,u)\equiv \lambda u-G(\lambda ,u)\) (for the nondegenerate case see \textit{H. Amann} [SIAM Rev. 18, 620-709 (1976; Zbl 0345.47044)]). In the sequel, under the so-called slope condition on the nonlinear term \(f(\lambda ,x,u)\equiv f(x,u)\), necessary and sufficient conditions for the existence of positive solutions to (1) are derived, by means of the super-subsolution method (for the nondegenerate case see \textit{H. Brezis} and \textit{L. Oswald} [Nonlinear Anal., Theory Methods Appl. 10, 55-64 (1986; Zbl 0593.35045)]). As consequence of the preceding results, the global bifurcation theory is studied in the following both cases: \(f(\lambda ,x,u)\equiv \lambda u-h(u)\) and \(f(\lambda ,x,u)\equiv \lambda g(u).\) Finally, six examples and figures are given illustrating the bifurcation point for (1) with respect to different nonlinearities \(f(\lambda ,x,u).\)
scientific article; zbMATH DE number 1216683