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

Giovanni Molica Bisci (Author), Dušan Repovš (Author), Luca Vilasi (Author)

Abstract

In this paper, we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian ▫$A_{1/2}$▫ in a smooth bounded domain ▫$\Omega \subset \mathbb{R}^n$▫ ▫$(n \ge 2)$▫ and with Dirichlet zero-boundary conditions, i.e. ▫$$ \begin{cases} A_{1/2}u = \lambda f(u) & \text{in} \quad \Omega \\ u = 0 & \text{on} \quad \partial \Omega. \end{cases}$$▫ The existence of at least three ▫$L^\infty$▫-bounded weak solutions is established for certain values of the parameter ▫$\lambda$▫ 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 method;multiple solutions;

Data

Language:	English
Year of publishing:	2017
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.95
COBISS:	17736793
ISSN:	0003-6811
Views:	511
Downloads:	341
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	str. 1483-1496
Volume:	ǂVol. ǂ96
Issue:	ǂno. ǂ9
Chronology:	2017
DOI:	10.1080/00036811.2016.1221069
ID:	11217776