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 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

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 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

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 (equations found) where ( 12\u394)sis the fractional Laplace operator, s 08 (0, 1), \u3a9 is an open bounded set of Rn, n > 2s, with Lipschitz boundary, \u3bb > 0 is a real parameter and 2 17 = 2n/(n 12 2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation (equations found), where LKis a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2 17 12 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 \u3bb1,sis the first eigenvalue of the non-local operator ( 12\u394)swith homogeneous Dirichlet boundary datum, then for any \u3bb 08 (0, \u3bb1,s) there exists a non-trivial solution of the above model equation, provided n 654s. 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

Fractional Laplacian equations with critical Sobolev exponent

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator LK (Formula Presented), that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445\u20132464, 2013). Here s 08(0,1),\u3a9 is an open bounded set of Rn,n>2s, with continuous boundary, \u3bb is a positive real parameter, 2 17=2n/(n-2s) is a fractional critical Sobolev exponent and f is a lower order perturbation of the critical power |u|2 17-2u, while LK is the integrodifferential operator defined as (Formula Presented). Under suitable growth condition on f, we show that this problem admits non-trivial solutions for any positive parameter \u3bb. This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when K(x)=|x|-(n+2s) (this gives rise to the fractional Laplace operator -(-\u394)s), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities

Variational methods for non-local operators of elliptic type

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

A Brezis-Nirenberg resul for non-local critical equations in low dimension

The present paper is devoted to the study of a nonlocal fractional equation involving critical nonlinearities. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when the set Omega is an open bounded subset of R-n with n >= 4s and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when 2s < n < 4s. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when s = 1 and consequently n = 3) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators

A multiplicity result for a class of nonlinear variational inequalities

We consider a class of variational inequalities and we give an existence result of a nonnegative, not identically zero solution. Such result generalizes the ones obtained by other authors through topological methods, to nonlinear variational inequalities. We also obtain the existence of at least two not identically zero solutions for a class of semilinear elliptic variational inequalities. Our proof of the existence result is based on the so called direct method, i.e., we introduce a suitable functional and we prove that it has a minimum, which is a solution of the variational inequality

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). In this paper we first study the problem in a general framework; indeed we consider an equation driven by a general non-local integrodifferential operator and in presence of a lower order perturbation of the critical power. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for an equation driven by the fractional Laplacian (-Delta)(s); that is, we show that if lambda(1,s) is the first eigenvalue of the non-local operator (-Delta)(s) with homogeneous Dirichlet boundary datum, then for any lambda is an element of (0, lambda(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

A Brezis-Nirenberg result for non-local critical equations in low dimension

The present paper is devoted to the study of the following nonlocal fractional equation involving critical nonlinearities { (-\u3b4) 08u -u = u2-2u in \u3c9 u = 0 in Rn n \u3c9 where s 2 (0; 1) is fixed, (-\u3b4)s is the fractional Laplace operator, is a positive parameter, 2 is the fractional critical Sobolev exponent and is an open bounded subset of Rn, n > 2s , with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when is an open bounded subset of Rn with n > 4s and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when 2s < n < 4s . In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when s = 1 (and consequently n = 3) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4] . In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators