The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. We prove this result for a general integrodifferential operator of fractional type and, as a particular case, we derive an existence theorem for the fractional Laplacian. As far as we know, all these results are new

In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator, depending on a real parameter and with the nonlinear term which satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem. As a particular case, we derive an existence theorem for an equation driven by the fractional Laplacian. Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators

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 equationwhere (−Δ) s is the fractional Laplace operator, s ∈ (0, 1), Ω is an open bounded set of R n , 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 equationwhere L K is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u| 2 * −2 u. 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 λ 1,s is the first eigenvalue of the non-local operator (−Δ) s with homogeneous Dirichlet boundary datum, then for any λ ∈ (0, λ 1,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.

Abstract. In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that iswhere ei , λi are the eigenfunctions and the eigenvalues of the Laplace operator −∆ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei . Aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

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 classical 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.