Nonlinear equations involving the square root of the Laplacian

Vincenzo Ambrosio (Author), Giovanni Molica Bisci (Author), Dušan Repovš (Author)

Abstract

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian ▫$A_{1/2}$▫ in a smooth bounded domain ▫$\Omega\subset \mathbb{R}^n$▫ (▫$n\geq 2$▫) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation ▫$$ \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. $$▫ The existence of at least two non-trivial ▫$L^{\infty}$▫-bounded weak solutions is established for large value of the parameter ▫$\lambda$▫ requiring that the nonlinear term ▫$f$▫ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.

Keywords

fractional Laplacian;variational methods;multiple solutions;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.951.6
COBISS:	18407513
ISSN:	1937-1632
Views:	520
Downloads:	170
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	str. 151-170
Volume:	ǂVol. ǂ12
Issue:	ǂiss. ǂ2
Chronology:	Apr. 2019
DOI:	10.3934/dcdss.2019011
ID:	11193865