A Bifurcation Result for Non-local Fractional Equations

In the present paper we consider problems modeled by the following non-local fractional equation { (−∆)u− λu = μf(x, u) in Ω u = 0 in R \ Ω , where s ∈ (0, 1) is fixed, (−∆) is the fractional Laplace operator, λ and μ are real parameters, Ω is an open bounded subset of R, n > 2s , with Lipschitz boundary and f is a function satisfying suitable regularity and growth conditions. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial and non-negative (non-positive) solution, provided the parameters λ and μ lie in a suitable range. The existence result obtained in the present paper may be seen as a bifurcation theorem, which extends some results, well known in the classical Laplace setting, to the non-local fractional framework.