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

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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.

Language:English
Keywords:fractional Laplacian, variational methods, multiple solutions
Work type:Article (dk_c)
Tipology: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
UDC:517.951.6
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
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Downloads:147
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Record is a part of a journal

Title:Discrete and continuous dynamical systems
Shortened title:Discret. contin. dyn. syst., Ser. S
Publisher:The American Institute of Mathematical Sciences
ISSN:1937-1632
COBISS.SI-ID:16098905 This link opens in a new window

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