Eigenvalues for systems of fractional $p$-Laplacians

Abstract

We study the eigenvalue problem for a system of fractional p−Laplacians, that is,We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues λn such that λn → ∞ as n → ∞.In addition, we study the limit as p → ∞ of the first eigenvalue, λ 1,p , andHere R(Ω) := max x∈Ω dist(x, ∂Ω) and [w]t,∞ := sup (x,y)∈Ω |w(y)−w(x)| |x−y| t . Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigen-pairs.

A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to\infty$

“…In our approach, it is more convenient to use for fixed values of p a notion of decoupled viscosity solution for (1.1), see [5] and [18] for the corresponding definitions in the local and nonlocal cases. Indeed, we consider the couple (u, v) as a viscosity solution for each equation of system (1.1), separately as follows:…”

“…Let us add that the same type of analysis has been carried out for systems driven by local or nonlocal p−Laplacians in [5] and [18], respectively.…”

“…We remark that the study of the aforementioned issues may give some insight on the connection between problems that admit distributional formulation with their limiting counterpart without this kind of structure. For a better comprehension on this subject we refer the reader to [5], [10], [11], [12], [13], [15], [16], [17], [18], [20], [22], [29] or [37].…”

“…On the other hand, if R > 1, the presence of the two fractional p−Laplacians and the limiting ratio Γ ∈ (0, 1) come into the play trough the convex combination Γr + (1 − Γ)s (cf. [18,Theorem 1.2]). The proof of such a dichotomy phenomenon strongly rely on a very specific choice of test functions in the limiting functional (1.11), see p. 17 below for details.…”

“…In order to establish our results, we have to overcome some technical obstacles and adopt certain alternative approaches, which, for the best of our knowledge, have not been put into practice for this kind of problems before (cf. [5], [18], [24], [29] and [34]), see Sections 2, 3, 5 and Appendix 6 for more details.…”