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We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, withs∈(0,1). These identities involve local boundary terms, in which the quantityplays the role that∂u∕∂νplays in the second-order case. Here,uis any solution toLu = f(x,u) in Ω, withu = 0 in ℝn∖Ω, anddis the distance to∂Ω. [ABSTRACT FROM AUTHOR]

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernelsKsatisfying thatK(y)|y|n+σis nondecreasing along rays from the origin, for some σ ∈ (0, 2) in caseaij ≡ 0 and for σ = 2 in case that (aij) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems forLin star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation tonand σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ)s(heres > 1) or the fractionalp-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin. [ABSTRACT FROM AUTHOR]

We consider the class of semi-stable positive solutions to semilinear equations − Δu = f(u) in a bounded domain Ω ⊂ ℝ n of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n − m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori L p and bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of − Δu = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 4 for any domain Ω, and in dimensions 5 ≤ n ≤ 9 in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities f in dimensions 5 ≤ n ≤ 9. [ABSTRACT FROM PUBLISHER]