On the fractional P–Q laplace operator with weights

Abstract

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $p\mathrm{\&q}$ Laplacian operator with indefinite weights ${(-{\mathrm{\Delta }}_p)}^{\alpha }u+{(-{\mathrm{\Delta }}_q)}^{\beta }u=\lambda [a{\left|u\right|}^{p-2}u+b{\left|u\right|}^{q-2}u]\mathrm{in}\mathrm{\Omega },$where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $\mathrm{\Omega }={\mathbb{R}}^N$ and $b\equiv 0$ a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding ${\mathcal{W}}^{\alpha ,p}\left({\mathbb{R}}^N\right)\hookrightarrow {\mathcal{W}}^{\beta ,q}\left({\mathbb{R}}^N\right)$ in ${\mathbb{R}}^N$, we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $p\mathrm{\&q}$ Laplacian problems in ${\mathbb{R}}^N$, Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $(p,q)$-Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems.

Keywords:

• Fractional p–q laplacian
• variational methods
• mountain pass theorem

AMS Classifications:

• 35J70
• 35J62
• 35B50
• 35B51

Acknowledgements

We would like to express our sincere thanks to the reviewer for their careful reading of the original manuscript and their helpful suggestions. We also deeply thank the editors for their correspondence.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by National key program for the development of Mathematics in the period from 2021 to 2030 (Grant B2024-CTT-01).