Some uniqueness results in quasilinear subhomogeneous problems

TL;DR

This paper proves uniqueness of certain solutions in quasilinear elliptic problems, including generalized p-Laplacian and (p,r)-Laplacian cases, using a new criterion and hidden convexity arguments.

Contribution

It introduces a new criterion for establishing uniqueness in quasilinear elliptic problems and applies it to generalized p-Laplacian and (p,r)-Laplacian operators.

Findings

Uniqueness among strongly positive solutions

Uniqueness among nonnegative global minimizers

Applicability to nonhomogeneous operators like (p,r)-Laplacian

Abstract

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called $(p,r)$-Laplacian are also treated.