Nonlinear equations involving the square root of the Laplacian
ID Ambrosio, Vincenzo (Author), ID Molica Bisci, Giovanni (Author), ID Repovš, Dušan (Author)

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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
Work type:Article
Typology:1.01 - Original Scientific Article
Organization:PEF - Faculty of Education
FMF - Faculty of Mathematics and Physics
Number of pages:Str. 151-170
Numbering:Vol. 12, iss. 2
PID:20.500.12556/RUL-108787 This link opens in a new window
ISSN on article:1937-1632
DOI:10.3934/dcdss.201901 This link opens in a new window
COBISS.SI-ID:18407513 This link opens in a new window
Publication date in RUL:25.07.2019
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Title:Discrete and continuous dynamical systems. Series S
Shortened title:Discret. contin. dyn. syst., Ser. S
Publisher:American Institute of Mathematical Sciences
COBISS.SI-ID:16098905 This link opens in a new window

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