POSITIVE SOLUTIONS FOR THE PERIODIC SCALAR $p$-LAPLACIAN: EXISTENCE AND UNIQUENESS

Abstract

We study a nonlinear periodic problem driven by the scalar $p$-Laplacian. The reaction term is a Carathéodory function $f(t,x)$ which satisfies only a unilateral growth condition in the $x$-variable. Assuming strict monotonicity for the quotient $f(t,x)\big/x^{p-1}$ and using variational methods coupled with suitable truncation techniques, we produce necessary and sufficient conditions for the existence and uniqueness of positive solutions.