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

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

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.

• Language: English
• Keywords: fractional Laplacian, variational method, multiple solutions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2017
• Number of pages: Str. 1483-1496
• Numbering: Vol. 96, no. 9
• PID: 20.500.12556/RUL-109675
• UDC: 517.95
• ISSN on article: 0003-6811
• DOI: 10.1080/00036811.2016.1221069
• COBISS.SI-ID: 17736793
• Publication date in RUL: 06.09.2019

MOLICA BISCI, Giovanni, REPOVŠ, Dušan and VILASI, Luca, 2017, Multiple solutions of nonlinear equations involving the square root of the Laplacian. Applicable analysis [online]. 2017. Vol. 96, no. 9, p. 1483–1496. [Accessed 14 June 2025]. DOI 10.1080/00036811.2016.1221069. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=109675

Record is a part of a journal

• Title: Applicable analysis
• Shortened title: Appl. anal.
• Publisher: Taylor & Francis
• ISSN: 0003-6811
• COBISS.SI-ID: 24981760