On Bobkov-Tanaka type spectrum for the double-phase operator

TL;DR

This paper extends the analysis of spectral properties to the double-phase operator, exploring conditions for positive solutions and employing advanced variational and inequality techniques.

Contribution

It introduces a Bobkov-Tanaka type spectrum framework for the double-phase operator, analyzing existence and non-existence regions for positive solutions.

Findings

Identifies parameter regions with positive solutions.

Establishes non-existence conditions.

Utilizes normalization, Nehari manifold, and Picone inequalities.

Abstract

Moving from the seminal papers by Bobkov and Tanaka \cite{BT,BT2,BT3} on the spectrum of the $(p,q)$-Laplacian, we analyze the case of the double-phase operator. We discuss the region of parameters in which existence and non-existence of positive solutions occur. The proofs are based on normalization procedures, the Nehari manifold, and truncation techniques, exploiting Picone-type inequalities and an ad-hoc strong maximum principle.