This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

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.

https://www.ams.org/tran/2015-367-01/S0002-9947-2014-05884-4/S0002-9947-2014-05884-4.pdf

The Brezis-Nirenberg result for the fractional Laplacian

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger–Kirchhoff type $$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$
 in $${\mathbb {R}}^N$$
 , where $$(-\varDelta )^s_p$$
 is the fractional p-Laplacian operator, with $$0<s<1<p<\infty $$
 and $$ps<N$$
 , the nonlinearity $$f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$
 is a Caratheodory function and satisfies the Ambrosetti–Rabinowitz condition, $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+$$
 is a potential function and $$g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$
 is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p -Laplacian in $${\mathbb {R}}^N$$ R N

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 bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

/pdf/on-the-spectrum-of-two-different-fractional-operators-coql44e4ce.pdf

On the spectrum of two different fractional operators

Weakly Differentiable Functions

In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in  Ω , u = 0 in  R n ∖ Ω where L K is an integrodifferential operator with kernel  K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

A critical Kirchhoff type problem involving a nonlocal operator

Abstract In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in  Ω , u = 0 in  R n ∖ Ω where L K is an integrodifferential operator with kernel  K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

A critical Kirchhoff type problem involving a non-local operator

Aim of this paper is to give the details of the proof of some density properties of smooth and compactly supported functions in the fractional Sobolev spaces and suitable modifica- tions of them, which have recently found application in variational problems. The arguments are rather technical, but, roughly speaking, they rely on a basic technique of convolution (which makes functions C 1 ), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.

/pdf/density-properties-for-fractional-sobolev-spaces-14q9ih9ct9.pdf

Density properties for fractional sobolev spaces

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problem 
−M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω,
where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

Alessio Fiscella

Papers

A critical Kirchhoff type problem involving a nonlocal operator

A critical Kirchhoff type problem involving a non-local operator

Density properties for fractional sobolev spaces

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity