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

Laura Gambera, Umberto Guarnotta

TL;DR

This work extends Bobkov–Tanaka-type spectrum concepts to the double-phase operator by studying the Dirichlet problem $- abla\cdot(a(x)|\nabla u|^{p-2}\nabla u)-\Delta_q u= \alpha a(x)|u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega$, with $u=0$ on $\partial\Omega$. It builds a variational framework in Musielak–Orlicz spaces, defines the energy $E_{\alpha,\beta}$ and its Nehari manifold, and introduces a linear independence condition between first eigenfunctions of the weighted $p$- and unweighted $q$-Laplacians, ${\rm (LI)}: \varphi_p^a \neq k\varphi_q$, to describe the spectrum. The paper proves a complete existence/nonexistence map for positive solutions, including a separating curve $\mathcal{C}=\{(\lambda^*(s)+s,\lambda^*(s))\,|\, s\in\mathbb{R}\}$ tied to the parameter $s=\alpha-\beta$, with detailed behavior depending on whether ${\rm (LI)}$ holds. The approach combines normalization, truncation, Picone-type inequalities, and a strong maximum principle to handle unbalanced growth, yielding sharp solvability criteria and shedding light on spectral structure for nonhomogeneous elliptic operators.

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.

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

TL;DR

This work extends Bobkov–Tanaka-type spectrum concepts to the double-phase operator by studying the Dirichlet problem in a bounded domain , with on . It builds a variational framework in Musielak–Orlicz spaces, defines the energy and its Nehari manifold, and introduces a linear independence condition between first eigenfunctions of the weighted - and unweighted -Laplacians, , to describe the spectrum. The paper proves a complete existence/nonexistence map for positive solutions, including a separating curve tied to the parameter , with detailed behavior depending on whether holds. The approach combines normalization, truncation, Picone-type inequalities, and a strong maximum principle to handle unbalanced growth, yielding sharp solvability criteria and shedding light on spectral structure for nonhomogeneous elliptic operators.

Abstract

Moving from the seminal papers by Bobkov and Tanaka \cite{BT,BT2,BT3} on the spectrum of the -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.
Paper Structure (3 sections, 12 theorems, 130 equations, 1 figure)

This paper contains 3 sections, 12 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. existence and non-existence results

Key Result

Proposition 2.2

The embedding $W^{1,q}_0(\Omega)\hookrightarrow L^\theta(\Omega)$ is compact.

Figures (1)

  • Figure 1: Summary of results. Existence of positive solutions is guaranteed in the white regions, while no positive solutions exist within the dotted regions. The curve $\mathcal{C}$ is represented with a black line, while the dotted oblique lines are related to the change of variable $s=\alpha-\beta$.

Theorems & Definitions (26)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 16 more