A concave—convex elliptic problem involving the fractional Laplacian

Abstract

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

Critical and subcritical fractional problems with vanishing potentials

“…Recently, Caffarelli and Silvestre [16] developed a local interpretation of the fractional Laplacian given in R N by considering a Neumann type operator in the extended domain R N +1 + defined by {(x, t) ∈ R N +1 : t > 0}. A similar extension, for nonlocal problems on bounded domain with the zero Dirichlet boundary condition, was established, for instance, by Cabrè and Tan in [15], Tan [37], Capella, Dàvila, Dupaigne and Sire [17], Brändle, Colorado, de Pablo and Sànchez [13]. It is worth noticing that, in a bounded domain, the Fourier definition of the fractional laplacian and its local Caffarelli-Silvestre interpretation do not agree, see the discussion developed [32] for more details.…”

Fractional Kirchhoff equation with a general critical nonlinearity

“…When a = 1 and b = 0, (1.1) becomes the fractional Schrödinger equations which have been studied by many authors. We refer the readers to [2,[5][6][7] and the references therein for the details. When s = 1, the problem (1.1) reduces to the well-known Kirchhoff equation…”

The Brezis–Nirenberg problem for nonlocal systems

“…We point out that we adopt in the paper the integral definition of the fractional laplacian in a bounded domain and we do not exploit any localization procedure based upon the Caffarelli-Silvestre extension [4], as done e.g. in [2]. See [14] for a nice comparison between these two different notions of fractional laplacian in bounded domains.…”

“…This is a contradiction with (2.7), because µ 2 < 0 and u, v > 0. Now we cover case (2). If A is the zero matrix, we getˆ∂ .…”