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

Abstract

Moving from the seminal papers by Bobkov and Tanaka [“On positive solutions for (p, q)-Laplace equations with two parameters,” Calc. Var. Partial Differ. Equ., vol. 54, pp. 3277–3301, 2015, “Remarks on minimizers for (p, q)-Laplace equations with two parameters,” Commun. Pure Appl. Anal., vol. 17, pp. 1219–1253, 2018, “Multiplicity of positive solutions for (p, q)-Laplace equations with two parameters,” Commun. Contemp. Math., vol. 24, 2022, Art. no. 2150008] 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.

Keywords

• double-phase operator
• non-homogeneous spectrum
• nehari manifold
• picone inequality
• 35j60
• 35j25
• 35b38
• 35p30