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"... Abstract. In this paper, we study equations driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. More precisely, we study the problem{ −LKu + V (x)u = |u|p−2u, in Ω, u = 0, in RN \ Ω, where 2 < p < 2∗s = 2N N−2s, Ω is an open bounded domain in R ..."

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Abstract. In this paper, we study equations driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. More precisely, we study the problem{ −LKu + V (x)u = |u|p−2u, in Ω, u = 0, in RN \ Ω, where 2 < p < 2∗s = 2N N−2s, Ω is an open bounded domain in R N for N> 2 and V is a L∞ potential such that −LK + V is positive definite. As a particular case, we study the problem{ (−∆)su + V (x)u = |u|p−2u, in Ω, u = 0, in RN \ Ω, where (−∆)s denotes the fractional Laplacian (with 0 < s < 1). We give assumptions on V, Ω and K such that ground state solutions (resp. least energy nodal solutions) respect the symmetries of some first (resp. second) eigenfunctions of −LK + V, at least for p close to 2. We study the uniqueness, up to a multiplicative factor, of those types of solutions. The results extend those obtained for the local case. 1.