Nonlinear equations involving the square root of the Laplacian

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

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.

• Jezik: Angleški jezik
• Ključne besede: fractional Laplacian, variational methods, multiple solutions
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: PEF - Pedagoška fakulteta; FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2019
• Št. strani: Str. 151-170
• Številčenje: Vol. 12, iss. 2
• PID: 20.500.12556/RUL-108787
• UDK: 517.951.6
• ISSN pri članku: 1937-1632
• DOI: 10.3934/dcdss.201901
• COBISS.SI-ID: 18407513
• Datum objave v RUL: 25.07.2019

AMBROSIO, Vincenzo, MOLICA BISCI, Giovanni in REPOVŠ, Dušan, 2019, Nonlinear equations involving the square root of the Laplacian. Discrete and continuous dynamical systems. Series S [na spletu]. 2019. Vol. 12, no. 2, p. 151–170. [Dostopano 22 avgust 2025]. DOI 10.3934/dcdss.201901. Pridobljeno s: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=slv&id=108787

Gradivo je del revije

• Naslov: Discrete and continuous dynamical systems. Series S
• Skrajšan naslov: Discret. contin. dyn. syst., Ser. S
• Založnik: American Institute of Mathematical Sciences
• ISSN: 1937-1632
• COBISS.SI-ID: 16098905