A sharp eigenvalue theorem for fractional elliptic equations

Abstract

By using variational methods, in this paper we study a nonlinear elliptic problem defined in a bounded domain Ω ⊂ ℝ N , with smooth boundary ∂Ω, involving fractional powers of the Laplacian operator together with a suitable nonlinear term f. More precisely, we prove a characterization theorem on the existence of one weak solution for the elliptic problem $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\alpha /2}}\mu = \lambda f(\mu )in\Omega ,} \\ {u > 0in\Omega ,} \\ {u = 0in\partial \Omega ,} \end{array}} \right.$$ , where α ∈ (0, 2), N > α, λ > 0 and (−Δ)α/2 denotes the nonlocal fractional Laplacian operator. Our result extends to the nonlocal setting recent theorems for ordinary and classical elliptic equations, as well as a characterization for elliptic problems on certain non-smooth domains. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.