Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

Abstract

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ* > 0 such that for λ < λ*, the problem has no positive solution; for λ = λ*, it has at least one positive solution; and for λ > λ*, it has at least two positive solutions.