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 the present paper, we study the following singular Kirchhoff problem M ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ( − Δ ) s u = λ f ( x ) u − γ + g ( x ) u 2 s ∗ − 1 in Ω , u > 0 in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is an open bounded domain, dimension N > 2 s with s ∈ ( 0 , 1 ) , 2 s ∗ = 2 N ∕ ( N − 2 s ) is the fractional critical Sobolev exponent, parameter λ > 0 , exponent γ ∈ ( 0 , 1 ) , M models a Kirchhoff coefficient, f ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is a positive weight, while g ∈ L ∞ ( Ω ) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ . This approach allows us to avoid any restriction on the boundary of Ω .

The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions.
These equations have a variational structure and we find a solution for them
using the Saddle Point Theorem. We prove this result for a general
integrodifferential operator of fractional type
and from this, as a particular case, one can derive an
existence theorem for the fractional Laplacian, finding 
solutions of the equation
$$
\begin{cases}
(-\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,\\
u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega,
\end{cases}
$$
where the nonlinear term $f$ satisfies a linear growth condition.

/pdf/saddle-point-solutions-for-non-local-elliptic-operators-1zfpo0oca8.pdf

Saddle point solutions for non-local elliptic operators

In this paper, we deal with a Kirchhoff type problem driven by a nonlocal fractional integrodifferential operator $\mathcal L_K$, that is, $$ -M( \|u \|^2)\mathcal L_Ku=\lambda f(x,u) \Big [\int_\Omega F(x,u(x))dx \Big ]^r+ \left |u \right |^{2^* -2}u \quad \mbox{in }\Omega, $$ $u=0$ in $ \mathbb{R}^{n}\setminus\Omega, $ where $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $M$ and $f$ are continuous functions, $ \left \|\cdot \right \|$ is a functional norm, $$ F(x,u(x))=\int^{u(x)}_0 f(x,\tau)d\tau , $$ $2^*$ is a fractional Sobolev exponent, $\lambda$ and $r$ are real parameters. For this problem, we prove the existence of infinitely many solutions, through a suitable truncation argument and exploiting the genus theory introduced by Krasnoselskii.

Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator

This paper deals with the existence, multiplicity and the asymptotic behavior of nontrivial solutions for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Hardy potential. In particular, we consider $$ \left\{ \begin{array}{ll} (- \Delta)^{s}u - \gamma \displaystyle \frac{u}{|x|^{2s}} = \lambda u + \theta f(x,u) +g(x,u) & \mbox{ in }\Omega,\\ u=0 & \mbox{in} \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N$ is a bounded domain, $\gamma, \lambda$ and $\theta$ are real parameters, the function $f$ is a subcritical nonlinearity, while $g$ could be either a critical term or a perturbation.

Alessio Fiscella

Papers

The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

Saddle point solutions for non-local elliptic operators

Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator

On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order