Counterexamples to some conjectures on the number of solutions of nonlinear equations
In this paper, we construct 3 counterexamples. Firstly, we construct an example which shows that a theorem of \textit{D. C. Clark} [Indiana Univ. Math. J. 22, 65-74 (1972; Zbl 0228.58006)] for nonlinear second order elliptic partial differential equations depends essentially on an oddness assumption. More precisely, we construct nonlinearities which cross many eigenvalues but the equation has only 4 nontrivial solutions. Secondly, we construct a related example which shows that a slight generalization of the Lazer-McKenna conjecture [see \textit{A. C. Lazer} and \textit{P. J. McKenna}, Proc. R. Soc. Edinb., Sect. A 95, 275-283 (1983; Zbl 0533.35037)] is false. The generalization consists in replacing the first eigenfunction by a function close to the first eigenfunction. Once again, we find examples where the nonlinearity crosses many eigenvalues but there are only 4 solutions. Thirdly, we construct competing species type models for two species with diffusion in which there are a large but finite number of solutions whatever the stability properties of the ''trivial'' solutions. Moreover, we find the exact number of solutions and their stability. The essential idea in all the examples is to perturb from degenerate situations.
nonlinear second order elliptic partial differential equations
scientific article; zbMATH DE number 3886359
scientific article