Energy–based localization and multiplicity of radially symmetric states for the stationary p–Laplace diffusion

ABSTRACT

It is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm.

COMMUNICATED BY:

• D. Repovš

KEYWORDS:

• p–Laplacian
• radial solution
• positive solution
• Harnack inequality
• energy–based localization
• multiple solutions

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35J20
• 35J60
• 34B15 Secondary: 35J15
• 35J25

Additional information

Funding

P. Pucci was partly supported by the Italian MIUR project titled Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM–GNAMPA Project 2018 denominated Problemi non lineari alle derivate parziali (Prot_U-UFMBAZ-2018-000384), and of the Fondo Ricerca di Base di Ateneo – Esercizio 2015 of the University of Perugia, named PDEs e Analisi Nonlineare, coordinator P. Pucci.