Runge property and multiplicity of solutions for elliptic equations in \({\mathbb{R}}^ N\) with a monotone nonlinearity
The author proves, that the solutions of \[ -\Delta u+g(u)=f\in L^ 1_{loc}({\mathbb{R}}^ n), \] where g is nondecreasing and Lipschitzcontinuous have the Runge property; that is, if \(\Omega\subset \Omega '\) are two simply connected domains, any solution u in \(\Omega\) can be uniformly approximated in compact subsets of \(\Omega\) by solutions in \(\Omega\) '. The proof relies on a homotopy method via the linearization \(-\Delta v+g'(u)v=0\), for which the Runge property is known to hold by works of \textit{P. D. Lax} [Commun. Pure Appl. Math. 9, 747-766 (1956; Zbl 0072.330)] and \textit{E. Heinz} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1, 1-12 (1955; Zbl 0067.075)]. As a consequence, \(-\Delta u+| u|^{\alpha -1}u=f\) with \(0<\alpha <1\) has infinitely many solutions in \(D'({\mathbb{R}}^ n)\).
scientific article; zbMATH DE number 3983711
scientific article