Raffaella Servadei

The Brezis-Nirenberg result for the fractional Laplacian

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation { (−Δ)su− λu = |u|2−2u in Ω, u = 0 in Rn \ Ω , where (−Δ)s is the fractional Laplace operator, s ∈ (0, 1), Ω is an open bounded set of Rn, n > 2s, with Lipschitz boundary, λ > 0 is a real parameter and 2∗ = 2n/(n− 2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation { LKu+ λu+ |u|2 −2u+ f(x, u) = 0 in Ω, u = 0 in Rn \ Ω , where LK is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2−2u. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if λ1,s is the first eigenvalue of the non-local operator (−Δ)s with homogeneous Dirichlet boundary datum, then for any λ ∈ (0, λ1,s) there exists a non-trivial solution of the above model equation, provided n 4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.

Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators

The purpose of this paper is to derive some Lewy-Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy-Stampacchia estimates in more classi- cal cases as well. In particular, we can recover the known estimates for the standard Laplacian, the p-Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy-Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new.

On the spectrum of two different fractional operators

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given by where c ( n, s ) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is, where e i , λ i are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while a i represents the projection of u on the direction e i . The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

Weak and viscosity solutions of the fractional Laplace equation

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s2 (0; 1) is a xed parameter, is a bounded, open subset of R n (n > 1) with C 2 -boundary, and ( ) s is the fractional Laplacian operator, that may be dened as

The Yamabe equation in a non-local setting

Abstract. Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator such that in , in , where , is an open bounded set of ℝn, , with Lipschitz boundary, λ is a positive real parameter, is a fractional critical Sobolev exponent, while is the non-local integrodifferential operator As a concrete example, we consider the case when , which gives rise to the fractional Laplace operator . In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any , provided and λ is different from the eigenvalues of . This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri and by Gazzola and Ruf for the classical Laplace equation with critical nonlinearities. In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators.

A bifurcation result for non-local fractional equations

In the present paper, we consider problems modeled by the following non-local fractional equation

A Resonance Problem for Non-Local Elliptic Operators

\left\{\begin{array}{@{}l@{\quad}l@{}}(-\Delta)^{s} u-\lambda u = \mu f(x,u) & {\rm in}\, \Omega,\\[4pt] u = 0 &{\rm in}\, {\mathbb R}^{n}{\setminus} \Omega,\end{array} \right.

On weak solutions for p-Laplacian equations with weights

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 ℝn, 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.