Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction

Abstract

We consider a generalized logistic equation driven by the Neumann p-Laplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value lambda _*>0 of the parameter, such that if lambda >lambda _*, the problem has at least two positive solutions, if lambda =lambda _*, the problem has at least one positive solution and it has no positive solution if lambda in (0,lambda _*). Finally, we show that for all lambda geqslant lambda _*, the problem has a smallest positive solution.