The Brezis-Nirenberg result for the fractional Laplacian

By Raffaella Servadei, Enrico Valdinoci. 2014

590 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bfbec90f67e80598ccd0cee7862aafaabed49523

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Citations in the corpus

The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation (2016, background): “Recently many people also studied the Brezis-Nirenberg problem for elliptic equation driven by the fractional Laplacian, this type of problem are nonlocal in nature and we may refer the readers to [6, 34, 35] and the references therein for a recent progress.”
Critical Nonlocal Systems with Concave-Convex Powers (2016, background): “On the other hand, the fractional problems for p = 2 have been investigated by many researchers, see, for example, [2, 6, 23] for the critical case and [11] for the fractional Kirchhoff type problem.”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Last but not least, fractional elliptic problems with critical growth, in a bounded domain, have been studied by some authors in the last years, see [6, 7, 18, 28, 39, 41] and references therein.”
On doubly nonlocal fractional elliptic equations (2015, methodology): “Indeed, the nonlocal analysis which we perform in this paper in order to use the Mountain Pass Theorem is quite general and it was successfully exploited for other goals in several recent contributions (see [16, 19, 20, 21, 23, 28, 30, 31, 32] and [15] for various properties on the fractional Sobole”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “6) and Proposition 21 in [47], we have that (6.”
A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝN (2016, background): “A Brézis–Nirenberg-type result for the nonlocal fractional Laplacian in bounded domainwith homogeneous Dirichlet boundary datum is given in [37] by variational techniques; see also [36] for further results.”
A critical fractional equation with concave-convex power nonlinearities (2013, methodology): “0 and ε > 0 small enough, .72) Furthermore, by [18, Lemma 3.8] (see also [41, Proposition 22]) it follows that t ε,λ ε ε .77) Indeed from (2.75) we get provided ε is small enough Moreover, the function ] .”
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, unknown): “For more related results, we refer the readers to [38,44] and the references therein.”