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

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 fractional Laplacian, variational methods, multiple solutions Article (dk_c) 1.01 - Original Scientific Article PEF - Faculty of EducationFMF - Faculty of Mathematics and Physics 2019 Str. 151-170 Vol. 12, iss. 2 517.951.6 1937-1632 10.3934/dcdss.201901 18407513 423 147 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

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Title: Discrete and continuous dynamical systems Discret. contin. dyn. syst., Ser. S The American Institute of Mathematical Sciences 1937-1632 16098905