A Brezis-Nirenberg type result for a nonlocal fractional operator
(2017) Journal of the London mathematical society — Vol. 95, n° 1, p. 73-93 (2016)
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- Abstract
- The aim of this paper is to deal with the nonlocal fractional counterpart of the Laplace equation involving critical nonlinearities studied by Brezis and Nirenberg. Namely, our model is the equation (-Δ)s pu = |u|p ∗ s -2u + λg(x, u) inΩ u = 0 in Rn \ Ω, where (-Δ)s p is the fractional p-Laplace operator, s ∈ (0, 1), Ω is an open bounded set of Rn, 2s ps < n, with smooth boundary, λ > 0 is a real parameter, p ∗ s := pn/(n - ps) is a fractional critical Sobolev exponent, and g is a subcritical nonlinearity. In this setting, through variational techniques, we prove the existence of one weak solution for the above problem provided that λ is sufficiently small. In addition, if the perturbation term g vanishes at the origin, a multiplicity result is established. Finally, we emphasize that the summability exponent p in the results presented here should be greater than or equal to 2.
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Mawhin, J., & Molica Bisci, G. (2017). A Brezis-Nirenberg type result for a nonlocal fractional operator. Journal of the London mathematical society, 95(1), 73-93. https://doi.org/10.1112/jlms.12009 (Original work published 2016)