Bifurcation properties of semilinear elliptic equations in \(\mathbb{R}^ n\)
The authors consider the semilinear equation \(\Delta u + \lambda f(x)(u + h(u)) = 0\), for \(x \in \mathbb{R}^ n\), \(n \geq 3\), where \(h : \mathbb{R} \to \mathbb{R}^ +\) is a bounded Hölder continuous function satisfying \(h(0) = 0\) and \(\xi^{-1} h(\xi) \to a > 0\) as \(\xi \to 0^ +\) and \(f : \mathbb{R}^ n \to \mathbb{R}\) is a positive asymptotically radial function for which \(\int_{\mathbb{R}^ n} f(x)dx < \infty\). They show the existence of a connected branch of positive solutions \(u\) satisfying \(| x |^{n-2} u(x) \to c > 0\) as \(| x | \to \infty\) which bifurcates from infinity at \(\lambda_ 1\), the principal eigenvalue of the linear equation \(\Delta u + \lambda f(x)u = 0\), and from the trivial solutions at \(\lambda_ 0 = \lambda_ 1/(a + 1)\). The techniques used here are based on [\textit{A. Ambrosetti} and \textit{P. Hess}, J. Math. Anal. Appl. 73, 411-422 (1980; Zbl 0433.35026)].
scientific article; zbMATH DE number 527356