1,720,968 research outputs found
A Brezis-Nirenberg result for non-local critical equations in low dimension
No full textThe present paper is devoted to the study of the following nonlocal fractional equation involving critical nonlinearities { (-δ) ∈u -u = u2-2u in ω u = 0 in Rn n ω where s 2 (0; 1) is fixed, (-δ)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
On the spectrum of two different fractional operators
No full textIn 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. © Royal Society of Edinburgh 2014
Fractional Laplacian equations with critical Sobolev exponent
No full textIn 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):2445–2464, 2013). Here s∈(0,1),Ω is an open bounded set of Rn,n>2s, with continuous boundary, λ is a positive real parameter, 2∗=2n/(n-2s) is a fractional critical Sobolev exponent and f is a lower order perturbation of the critical power |u|2∗-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 λ. 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 -(-Δ)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
Periodic solutions of nonlinear impulsive differential inclusions with constraints
No full textIn this paper we obtain the existence of periodic solutions for nonlinear invariance problems monitored by impulsive differential inclusions subject to impulse effects. In our proof we use the result due to Hristova and Bainov concerning the existence of a periodic solution for impulsive differential equations, together with an approximation argument. Our Theorems 3.1 and 3.2 extend the existence result proved by Watson (see Remarks 3.1 and 3.2) and, moreover, improve the Hristova - Bainov existence theorem in the
case of "invariance" problems involving impulsive differential equations
A resonance problem for non-local elliptic operators
No full textIn this paper we consider a resonance problem driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation {(-δ)su=λa(x)u + f(x; u) in u = 0 in Rn n ; when λ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s 2 (0; 1) is fixed, is an open bounded set of Rn, n > 2s, with Lipschitz boundary, a is a Lipschitz continuous function, while f is a suffciently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator delta;
Variational methods for non-local operators of elliptic type
No full textIn this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem {LKu + λu + f(x, u) = 0 in Ω u = 0 in Rn\Ω, where λ is a real parameter and the nonlinear term f 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 Jλ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when λ < λ1 and λ ≥ λ1, where λ1 denotes the first eigenvalue of the operator-LK. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian {(-δ)su - λu = f(x, u) in Ω u = 0 in Rn\Ω. Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators
Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators
No full textThe 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
No full textThe 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 (−Δ)sis 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 (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 ∗ − 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,sis the first eigenvalue of the non-local operator (−Δ)swith 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
Weak and viscosity solutions of the fractional Laplace equation
Full text linkAim of this paper is to give a regularity result for weak solutions of a fractional Laplacian equation. In order to get this result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem.
As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of the fractional Laplacian is strictly positive
Mountain Pass solutions for non-local elliptic operators
No full textThe 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
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