The topological contribution of the critical points at infinity for critical fractional Yamabe-type equations
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ORIGINAL PAPER
The topological contribution of the critical points at infinity for critical fractional Yamabe-type equations Khadijah Abdullah Sharaf1 • Hichem Chtioui2 Received: 9 November 2019 / Accepted: 11 March 2020 Springer Nature Switzerland AG 2020
Abstract nþ2s In this paper, we study the critical fractional nonlinear PDE: ðDÞs u ¼ un2s , u [ 0 in X and u ¼ 0 on oX, where X is a thin annuli-domain of Rn ; n 2: We compute the evaluation of the difference of topology induced by the critical points at infinity between the level sets of the associated variational function. Our Theorem can be seen as a nonlocal analog of the result of Ahmedou and El Mehdi (Duke Math J 94:215–229, 1998) on the classical Yamabe-type equation. Keywords Fractional operator Calculus of variational Critical points at infinity Mathematics Subject Classification 35J65 58J20 58C30
1 Introduction In this paper, we consider the following critical fractional nonlinear PDE: 8 nþ2s s > < ðDÞ u ¼ un2s ; > :
u [ 0 in X; u ¼ 0 on oX:
ð1:1Þ
Here ðDÞs ; s 2 ð0; 1Þ, is the fractional Dirichlet Laplacian operator defined by using the spectrum of the Laplacian D in X with zero Dirichlet boundary condition, and X is a bounded regular domain of Rn ; n 2.
‘‘This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira.’’ & Hichem Chtioui [email protected] Khadijah Abdullah Sharaf [email protected] 1
Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Kingdom of Saudi Arabia
2
Department of Mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia SN Partial Differential Equations and Applications
11 Page 2 of 26
SN Partial Differ. Equ. Appl. (2020)1:11
In the last decades, partial differential equations involving ðDÞs have attracted the attention of several authors, this is essentially due to its numerous applications in various domains as biology, astrophysics, water waves, nonlocal diffusion, optics and medicine. The nonlocal character of ðDÞs presents mathematical difficulties. The breakthrough paper of Caffarelli and Silvester [12] has unlocked the situation and opened the door to other significant contributions, see for example [2, 4–6, 13, 15–20, 22]. Problem (1.1) has a variational structure, the solutions are the critical points of J; the Euler–Lagrange functional associated to (1.1). Let us first point out that the subcritical nþ2s problem of (1.1) obtained by modifying the exponent nþ2s n2s to n2s e, e [ 0. In this case, an existence result was obtained by Cabre´ and Tan [11]. They were able to prove that the Palais–Smale condition holds for the associated function. Such a property is not satisfied for (1.1). This leads to the possibility of existence of non-compact orbits of ðoJÞ along which J is bounded and its gradient tends to zero, the so-called critical points at infinity, see [7]. Although some results were established on (1.1), see for example [3, 25], questions concerning existence of solutions, m
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