Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

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The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

Introduction to Functional Analysis.

Mountain Pass solutions for non-local elliptic operators

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

In this paper we study the existence of non-trivial solutions for
equations driven by a non-local integrodifferential
operator $\mathcal L_K$ with homogeneous Dirichlet boundary
conditions. More precisely, we consider the problem
$$ \left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\
u=0  in \mathbb{R}^n \backslash Ω ,
\end{array} \right.
$$
where $\lambda$ 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
$\mathcal J_\lambda$ associated to the problem. Here we get such
critical points using both the Mountain Pass Theorem and the Linking
Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq
\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the
operator $-\mathcal L_K$.

As a particular case, we derive an existence theorem for the
following equation driven by the fractional Laplacian
$$ \left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) in Ω \\
u=0  in \mathbb{R}^n \backslash Ω.
\end{array} \right.
$$
Thus, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.

Variational methods for non-local operators of elliptic type

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 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

Raffaella Servadei

Papers

Mountain Pass solutions for non-local elliptic operators

Variational Methods for Nonlocal Fractional Problems

Variational methods for non-local operators of elliptic type

The Brezis-Nirenberg result for the fractional Laplacian

On the spectrum of two different fractional operators