A bifurcation result for non-local fractional equations

Abstract

In the present paper, we consider problems modeled by the following non-local fractional equation [Formula: see text] where s ∈ (0, 1) is fixed, (-Δ) s is the fractional Laplace operator, λ and μ are real parameters, Ω is an open bounded subset of ℝ n , 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.