Allegretto, Walter; Nistri, Paolo Elliptic equations with discontinuous nonlinearities. (English) Zbl 0827.35134 Topol. Methods Nonlinear Anal. 2, No. 2, 233-251 (1993). The paper deals with the existence and nonexistence of positive solutions of nonlinear elliptic equations, with the nonlinear term discontinuous in the unknown function. The authors distinguish between 3 types of solutions. Topological methods – specifically the degree theory – are employed to obtain the existence of solutions, either in \(\mathbb{R}^n\) or in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\). When the nonlinearity is of sublinear form, the existence of a positive type I solution is proved. The existence of a positive type II solution is proved both for superlinear and sublinear (under an additional condition) nonlinearity. The nonexistence is proved in sublinear case with small enough range of discontinuity. The question of the existence of the solution of type I for general superlinear case is left open. Reviewer: P.Burda (Praha) Cited in 4 Documents MSC: 35R05 PDEs with low regular coefficients and/or low regular data 35J60 Nonlinear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:existence; nonexistence; positive solutions; superlinear; sublinear × Cite Format Result Cite Review PDF Full Text: DOI