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Minimax Theory and its Applications 02 (2017), No. 2, 265--283
Copyright Heldermann Verlag 2017

Bounded Solutions to Nonlinear Problems in RN Involving the Fractional Laplacian Depending on Parameters

Said El Manouni
Dept. of Mathematics and Statistics, Faculty of Sciences, Al Imam Mohammad Ibn Saud Islamic University, P. O. Box 90950, Riyadh 11623, Saudi Arabia

Hichem Hajaiej
New York University, 1555 Centry Avenue, Pudong New District, Shanghai 200122, P. R. China

Patrick Winkert
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator (-Δ)s and nonlinearities that have subcritical growth. In the second part, based on a variational principle of B. Ricceri [A further critical points theorem, Nonlinear Anal. 71 (2009) 4151--4157], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.

Keywords: Fractional Laplacian, nonlocal eigenvalue problems, unbounded domains, existence and regularity, multiplicity results, Ricceri's principle.

MSC: 35R11, 35J20, 35J60, 46E35

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