The Brezis-Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains

In this paper we study the existence of nontrivial solutions to the well-known Brezis- Nirenb erg problem involving the fractional p-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincare<acute accent> inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue lambda(p,s)((sic)) with respect to the domain (sic). Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi-Perera-Squassina-Yang [The Brezis-Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis-Nirenb erg type results of Ramos-Wang-Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192-199] to the fractional p-Laplacian setting.