Gandal, SomnathSomnathGandalTyagi, JagmohanJagmohanTyagi2025-08-312025-08-312023-11-0110.1016/j.bulsci.2023.1033222-s2.0-85172221235https://d8.irins.org/handle/IITG2025/26570We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: {(−Δ)^su+λu=|u|^p−1uinΩ,N<inf>s</inf>u(x)=0inRⁿ∖Ω‾,u≥0inΩ, where λ>0 is a constant and Ω⊂Rⁿ is a bounded domain with smooth boundary. Here, [Formula presented], s∈(0,1). Due to the critical exponent in the problem, the corresponding functional J<inf>λ</inf> does not satisfy the Palais-Smale (PS)-condition and therefore one cannot use standard variational methods to find the critical points of J<inf>λ</inf>. We overcome such difficulties by establishing a bound for Rayleigh quotient and with the aid of nonlocal version of the Cherrier's optimal Sobolev inequality in bounded domains. We also show the uniqueness of these solutions in small domains.falseExistence and uniqueness | Fractional Laplacian | Positive solutions | Semilinear Neumann problemArticleNovember 20231103322arJournal0WOS:001073969600001