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Minimax Theory and its Applications 04 (2019), No. 1, 033--054
Copyright Heldermann Verlag 2019

Perturbed Problems Involving the Square Root of the Laplacian

Rossella Bartolo
Dip. di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy

Eduardo Colorado
Dep. de Matemáticas, Universidad Carlos III, Avda. Universidad 30, 28911 Leganés - Madrid, Spain
and: Instituto de Ciencias Matemáticas, C/Nicolás Cabrera 15, 28049 Madrid, Spain

Giovanni Molica Bisci
Dipartimento P.A.U., Univ. degli Studi Mediterranea, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy


\newcommand{\R}{{\mathbb R}} We prove multiplicity of solutions for perturbed problems involving the square root of the Laplacian $\mathcal{A}=(-\Delta)^{1/2}$. More precisely, we consider the problem \begin{equation*} \left\{ \begin{array}{ll} \mathcal{A}u=\lambda u + f(x,u) +\varepsilon g(x,u)& \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega, \end{array}\right. \end{equation*} where $\Omega\subset\R^N$ is a bounded domain, $\varepsilon\in\R$, $N>1$, $f$ is a subcritical function with asymptotic linear behavior at infinity, and $g$ is a continuous function. We also show the invariance under small perturbations of the number of distinct critical levels of the associated energy functional to the unperturbed problem, in both resonant and non-resonant case.

Keywords: Fractional Laplacian, variational methods, multiplicity of solutions.

MSC: 49J35, 35A15, 35S15.

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