Multiple solutions of nonlinear equations involving the square root of the Laplacian

Abstract

In this paper, we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian in a smooth bounded domain () and with Dirichlet zero-boundary conditions, i.e.

The existence of at least three -bounded weak solutions is established for certain values of the parameter requiring that the nonlinear term f is continuous and with a suitable growth. Our approach is based on variational arguments and a variant of Caffarelli–Silvestre’s extension method.

Keywords:

• Fractional Laplacian
• variational methods
• multiple solutions

AMS Subject Classifications:

• Primary: 49J35
• 35A15
• 35S15
• Secondary: 47G20
• 45G05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The paper has been carried out under the auspices of the INdAM - GNAMPA Project 2016 titled: Problemi variazionali su varietà Riemanniane e gruppi di Carnot and the Slovenian Research Agency [grant number P1-0292-0101], [grant number J1-7025-0101], [grant number J1-6721-0101], [grant number J1-5435-0101].