Nonlinear equations involving the square root of the Laplacian

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

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.

• Language: English
• Keywords: fractional Laplacian, variational methods, multiple solutions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 151-170
• Numbering: Vol. 12, iss. 2
• PID: 20.500.12556/RUL-108787
• UDC: 517.951.6
• ISSN on article: 1937-1632
• DOI: 10.3934/dcdss.201901
• COBISS.SI-ID: 18407513
• Publication date in RUL: 25.07.2019

AMBROSIO, Vincenzo, MOLICA BISCI, Giovanni and REPOVŠ, Dušan, 2019, Nonlinear equations involving the square root of the Laplacian. Discrete and continuous dynamical systems. Series S [online]. 2019. Vol. 12, no. 2, p. 151–170. [Accessed 14 June 2025]. DOI 10.3934/dcdss.201901. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=108787

Record is a part of a journal

• Title: Discrete and continuous dynamical systems. Series S
• Shortened title: Discret. contin. dyn. syst., Ser. S
• Publisher: American Institute of Mathematical Sciences
• ISSN: 1937-1632
• COBISS.SI-ID: 16098905