Positive solutions for generalized nonlinear logistic equations of superdiffusive type

Abstract

We consider a generalized version of the $p$-logistic equation. Using variational methods based on the critical point theory and truncation techniques, we prove a bifurcation-type theorem for the equation. So, we show that there is a critical value $\lambda_*> 0$ of the parameter $\lambda> 0$ such that the following holds: if $\lambda> \lambda_*$, then the problem has two positive solutions; if $\lambda=\lambda_*$, then there is a positive solution; and finally, if $0< \lambda< \lambda_*$, then there are no positive solutions.