# Double-phase problems and a discontinuity property of the spectrum

## TL;DR

This paper investigates a nonlinear eigenvalue problem involving the sum of p- and q-Laplacians, demonstrating the spectrum's continuity and revealing a discontinuity as a parameter approaches a critical value.

## Contribution

It establishes the continuity of the spectrum for the nonlinear eigenvalue problem and uncovers a discontinuity property related to the parametric (p,q)-differential operator.

## Key findings

- The spectrum of the problem is continuous.
- A discontinuity occurs in the spectrum as the parameter approaches 1 from below.
- The study enhances understanding of spectral properties of nonlinear differential operators.

## Abstract

We consider a nonlinear eigenvalue problem driven by the sum of $p$ and $q$-Laplacian. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ($p,q$)-differential operator as the parameter $\beta\rightarrow 1^-$.